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Properties

Label 4-6650e2-1.1-c1e2-0-10
Degree $4$
Conductor $44222500$
Sign $1$
Analytic cond. $2819.66$
Root an. cond. $7.28701$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 3·4-s − 2·6-s + 2·7-s + 4·8-s + 2·9-s + 3·11-s − 3·12-s + 2·13-s + 4·14-s + 5·16-s + 8·17-s + 4·18-s + 2·19-s − 2·21-s + 6·22-s + 6·23-s − 4·24-s + 4·26-s − 6·27-s + 6·28-s + 5·29-s + 20·31-s + 6·32-s − 3·33-s + 16·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s + 0.755·7-s + 1.41·8-s + 2/3·9-s + 0.904·11-s − 0.866·12-s + 0.554·13-s + 1.06·14-s + 5/4·16-s + 1.94·17-s + 0.942·18-s + 0.458·19-s − 0.436·21-s + 1.27·22-s + 1.25·23-s − 0.816·24-s + 0.784·26-s − 1.15·27-s + 1.13·28-s + 0.928·29-s + 3.59·31-s + 1.06·32-s − 0.522·33-s + 2.74·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44222500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44222500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44222500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2819.66\)
Root analytic conductor: \(7.28701\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44222500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(13.69215466\)
\(L(\frac12)\) \(\approx\) \(13.69215466\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_ab
11$D_{4}$ \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_r
13$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_ac
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
23$D_{4}$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_ba
29$D_{4}$ \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.29.af_cf
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
37$D_{4}$ \( 1 + 3 T + 69 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.37.d_cr
41$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_t
43$D_{4}$ \( 1 + 15 T + 135 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.43.p_ff
47$D_{4}$ \( 1 - 15 T + 143 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.47.ap_fn
53$D_{4}$ \( 1 - 5 T + 105 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.53.af_eb
59$D_{4}$ \( 1 - 7 T + 65 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.59.ah_cn
61$D_{4}$ \( 1 + 7 T + 127 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_ex
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$D_{4}$ \( 1 - 3 T + 137 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.71.ad_fh
73$D_{4}$ \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_eo
79$D_{4}$ \( 1 - 9 T + 171 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.79.aj_gp
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.83.q_iw
89$D_{4}$ \( 1 - 21 T + 223 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.89.av_ip
97$D_{4}$ \( 1 + 23 T + 319 T^{2} + 23 p T^{3} + p^{2} T^{4} \) 2.97.x_mh
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.113343079766978829306037240377, −7.76397103857842122376503768667, −7.26515288338486991701607517877, −6.89085955601426154330644495874, −6.71069658309639128444542539697, −6.46267136586649992525701313879, −5.82694045859651092717432059911, −5.68238933006562343198252388575, −5.30861876701184362521350162564, −4.95677603847335427578873781284, −4.61751584517881075365213618552, −4.27982127580356801820667820816, −3.86315597688686468380941173212, −3.54014079387177472120118257947, −2.94407591002727107010285089950, −2.84238859130347326586010538292, −2.12818602102218606984495146267, −1.41891201296721878789661852520, −1.13746484704639634510084703176, −0.901922748692216753912505284998, 0.901922748692216753912505284998, 1.13746484704639634510084703176, 1.41891201296721878789661852520, 2.12818602102218606984495146267, 2.84238859130347326586010538292, 2.94407591002727107010285089950, 3.54014079387177472120118257947, 3.86315597688686468380941173212, 4.27982127580356801820667820816, 4.61751584517881075365213618552, 4.95677603847335427578873781284, 5.30861876701184362521350162564, 5.68238933006562343198252388575, 5.82694045859651092717432059911, 6.46267136586649992525701313879, 6.71069658309639128444542539697, 6.89085955601426154330644495874, 7.26515288338486991701607517877, 7.76397103857842122376503768667, 8.113343079766978829306037240377

Graph of the $Z$-function along the critical line