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Properties

Label 4-35e4-1.1-c1e2-0-5
Degree $4$
Conductor $1500625$
Sign $1$
Analytic cond. $95.6811$
Root an. cond. $3.12756$
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-s + 2·4-s + 5·8-s − 6·9-s − 4·11-s + 5·16-s − 6·18-s − 4·22-s + 8·23-s − 2·29-s + 10·32-s − 12·36-s − 6·37-s − 12·43-s − 8·44-s + 8·46-s + 20·53-s − 2·58-s + 17·64-s + 4·67-s − 16·71-s − 30·72-s − 6·74-s − 8·79-s + 27·81-s − 12·86-s − 20·88-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s + 1.76·8-s − 2·9-s − 1.20·11-s + 5/4·16-s − 1.41·18-s − 0.852·22-s + 1.66·23-s − 0.371·29-s + 1.76·32-s − 2·36-s − 0.986·37-s − 1.82·43-s − 1.20·44-s + 1.17·46-s + 2.74·53-s − 0.262·58-s + 17/8·64-s + 0.488·67-s − 1.89·71-s − 3.53·72-s − 0.697·74-s − 0.900·79-s + 3·81-s − 1.29·86-s − 2.13·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1500625\)    =    \(5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(95.6811\)
Root analytic conductor: \(3.12756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1500625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.990155320\)
\(L(\frac12)\) \(\approx\) \(2.990155320\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_ab
3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_f
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bp
29$C_2^2$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_az
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_ab
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_dx
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2^2$ \( 1 + 16 T + 185 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.71.q_hd
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−10.01959184322465980084712559146, −9.699600750910145630848530593743, −8.806359948205660945161948872373, −8.641272752858889519330169278661, −8.428496916064441131782653601895, −7.70283930173604069466681300658, −7.50444860976862356443729336925, −6.92726841305019761257651557534, −6.74953425220276383926801424219, −6.01411625363100731726211490944, −5.54831764284406831054848586852, −5.39576834090927669355728915322, −4.86967648718154815102593211128, −4.50964608075976483409271282677, −3.68350998539542172005763398597, −3.16583587986373159734214660538, −2.85543615097272829993642620410, −2.28095635488370275222297630382, −1.73895422661891684138330226755, −0.62722834978033193815539962648, 0.62722834978033193815539962648, 1.73895422661891684138330226755, 2.28095635488370275222297630382, 2.85543615097272829993642620410, 3.16583587986373159734214660538, 3.68350998539542172005763398597, 4.50964608075976483409271282677, 4.86967648718154815102593211128, 5.39576834090927669355728915322, 5.54831764284406831054848586852, 6.01411625363100731726211490944, 6.74953425220276383926801424219, 6.92726841305019761257651557534, 7.50444860976862356443729336925, 7.70283930173604069466681300658, 8.428496916064441131782653601895, 8.641272752858889519330169278661, 8.806359948205660945161948872373, 9.699600750910145630848530593743, 10.01959184322465980084712559146

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