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Properties

Label 2-1904-476.135-c0-0-2
Degree $2$
Conductor $1904$
Sign $0.947 - 0.319i$
Analytic cond. $0.950219$
Root an. cond. $0.974792$
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  + (0.258 + 0.448i)3-s + (0.707 − 0.707i)7-s + (0.366 − 0.633i)9-s + (0.965 + 1.67i)11-s − 1.73·13-s + (0.5 + 0.866i)17-s + (0.5 + 0.133i)21-s + (0.707 − 1.22i)23-s + (−0.5 − 0.866i)25-s + 0.896·27-s + (0.707 + 1.22i)31-s + (−0.500 + 0.866i)33-s + (−0.448 − 0.776i)39-s − 1.00i·49-s + (−0.258 + 0.448i)51-s + ⋯
L(s)  = 1  + (0.258 + 0.448i)3-s + (0.707 − 0.707i)7-s + (0.366 − 0.633i)9-s + (0.965 + 1.67i)11-s − 1.73·13-s + (0.5 + 0.866i)17-s + (0.5 + 0.133i)21-s + (0.707 − 1.22i)23-s + (−0.5 − 0.866i)25-s + 0.896·27-s + (0.707 + 1.22i)31-s + (−0.500 + 0.866i)33-s + (−0.448 − 0.776i)39-s − 1.00i·49-s + (−0.258 + 0.448i)51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1904\)    =    \(2^{4} \cdot 7 \cdot 17\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(0.950219\)
Root analytic conductor: \(0.974792\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1904} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1904,\ (\ :0),\ 0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.407230799\)
\(L(\frac12)\) \(\approx\) \(1.407230799\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.73T + T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 0.517T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.740248162771570853240250596059, −8.700673569528244637719563459587, −7.87001746568159633302244459892, −6.95538959104009455157782318602, −6.64714787118121580465943028301, −5.03154550080646598404801402229, −4.49014095554876158102073627022, −3.88443343824757777748563736116, −2.52242672025114502018330956210, −1.38916341217935043774936476899, 1.29745561833815317456190818249, 2.42700677316046928072194148738, 3.28201020192719296336047208311, 4.60927433503944038424059727621, 5.34657057497909776470652159709, 6.09387281382190571911162767529, 7.35769986456912140334666104618, 7.62067515437959437802463736576, 8.568441108228104584382883085086, 9.286464979299899153861114562677

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