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Properties

Label 4-1220000-1.1-c1e2-0-5
Degree $4$
Conductor $1220000$
Sign $-1$
Analytic cond. $77.7882$
Root an. cond. $2.96980$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 5·9-s + 2·13-s + 16-s − 6·17-s + 5·18-s − 2·26-s − 32-s + 6·34-s − 5·36-s − 6·37-s + 4·41-s + 5·49-s + 2·52-s − 3·53-s + 8·61-s + 64-s − 6·68-s + 5·72-s − 3·73-s + 6·74-s + 16·81-s − 4·82-s + 15·89-s − 97-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 5/3·9-s + 0.554·13-s + 1/4·16-s − 1.45·17-s + 1.17·18-s − 0.392·26-s − 0.176·32-s + 1.02·34-s − 5/6·36-s − 0.986·37-s + 0.624·41-s + 5/7·49-s + 0.277·52-s − 0.412·53-s + 1.02·61-s + 1/8·64-s − 0.727·68-s + 0.589·72-s − 0.351·73-s + 0.697·74-s + 16/9·81-s − 0.441·82-s + 1.58·89-s − 0.101·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1220000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1220000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1220000\)    =    \(2^{5} \cdot 5^{4} \cdot 61\)
Sign: $-1$
Analytic conductor: \(77.7882\)
Root analytic conductor: \(2.96980\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1220000,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5 \( 1 \)
61$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 7 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.ac_c
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.g_s
19$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.19.a_abg
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.a_abj
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.31.a_i
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.g_cg
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.ae_ao
43$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.43.a_be
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.47.a_ak
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.d_ca
59$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.59.a_aw
67$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.67.a_dm
71$C_2^2$ \( 1 - 72 T^{2} + p^{2} T^{4} \) 2.71.a_acu
73$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.d_fm
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.83.a_cs
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.89.ap_iu
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.b_bm
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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

−7.88713410606036734952344793042, −7.41875765418937052134327717389, −6.97722194245656799479471415358, −6.46263841077439735764717258195, −6.08638513460703152167995485883, −5.78503129155512653732462171939, −5.16825815721256337110981670806, −4.74093668651875347469424049789, −4.05779300570008132450851922052, −3.46617854501668295847895907085, −3.03131079421991742586705847213, −2.30874140983669766043679977641, −2.01078346838779193065655282775, −0.888588884432719060629012098436, 0, 0.888588884432719060629012098436, 2.01078346838779193065655282775, 2.30874140983669766043679977641, 3.03131079421991742586705847213, 3.46617854501668295847895907085, 4.05779300570008132450851922052, 4.74093668651875347469424049789, 5.16825815721256337110981670806, 5.78503129155512653732462171939, 6.08638513460703152167995485883, 6.46263841077439735764717258195, 6.97722194245656799479471415358, 7.41875765418937052134327717389, 7.88713410606036734952344793042

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