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Properties

Label 2-3400-3400.3331-c0-0-1
Degree $2$
Conductor $3400$
Sign $-0.535 + 0.844i$
Analytic cond. $1.69682$
Root an. cond. $1.30262$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + i·5-s − 1.90·7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.951 − 0.309i)10-s + (0.587 + 1.80i)14-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s − 0.999·18-s + (−1.30 − 0.951i)19-s + (−0.587 − 0.809i)20-s + (0.363 + 1.11i)23-s − 25-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + i·5-s − 1.90·7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.951 − 0.309i)10-s + (0.587 + 1.80i)14-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s − 0.999·18-s + (−1.30 − 0.951i)19-s + (−0.587 − 0.809i)20-s + (0.363 + 1.11i)23-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $-0.535 + 0.844i$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (3331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ -0.535 + 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5357796501\)
\(L(\frac12)\) \(\approx\) \(0.5357796501\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 - iT \)
17 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + 1.90T + T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.045871251979170426105277912796, −7.68400066305056995001109605805, −7.13446083482565865097647039914, −6.29500443225531379541819096203, −5.74067767515348663471289665715, −4.12306843714713395955190449918, −3.61475953412610811394066714603, −3.01716319236378036064719094167, −2.10748875690340312301105403814, −0.40793945897465995556139615802, 1.10701117587395668348655884577, 2.60217845206672602408993503302, 3.88240879809337846934769757714, 4.53453240913539254345915703794, 5.48867144110043654670026074320, 6.04919915036573101466043933532, 6.82395221119838679907313712184, 7.53422262209496928313685427205, 8.344511960359188169001630080916, 8.918685490191186165140757300669

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