Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 59 $
Sign $-0.652 - 0.757i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−1.65 + 0.509i)3-s + 4-s − 2.90i·5-s + (1.65 − 0.509i)6-s − 3.06·7-s − 8-s + (2.48 − 1.68i)9-s + 2.90i·10-s + 1.35·11-s + (−1.65 + 0.509i)12-s + 7.06i·13-s + 3.06·14-s + (1.48 + 4.80i)15-s + 16-s − 0.568i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.955 + 0.294i)3-s + 0.5·4-s − 1.29i·5-s + (0.675 − 0.208i)6-s − 1.16·7-s − 0.353·8-s + (0.826 − 0.562i)9-s + 0.917i·10-s + 0.408·11-s + (−0.477 + 0.147i)12-s + 1.95i·13-s + 0.820·14-s + (0.382 + 1.24i)15-s + 0.250·16-s − 0.137i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(354\)    =    \(2 \cdot 3 \cdot 59\)
\( \varepsilon \)  =  $-0.652 - 0.757i$
motivic weight  =  \(1\)
character  :  $\chi_{354} (353, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 354,\ (\ :1/2),\ -0.652 - 0.757i)$
$L(1)$  $\approx$  $0.0771695 + 0.168225i$
$L(\frac12)$  $\approx$  $0.0771695 + 0.168225i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;59\}$, \(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.65 - 0.509i)T \)
59 \( 1 + (3.07 + 7.03i)T \)
good5 \( 1 + 2.90iT - 5T^{2} \)
7 \( 1 + 3.06T + 7T^{2} \)
11 \( 1 - 1.35T + 11T^{2} \)
13 \( 1 - 7.06iT - 13T^{2} \)
17 \( 1 + 0.568iT - 17T^{2} \)
19 \( 1 + 4.42T + 19T^{2} \)
23 \( 1 + 7.20T + 23T^{2} \)
29 \( 1 - 2.41iT - 29T^{2} \)
31 \( 1 - 6.44iT - 31T^{2} \)
37 \( 1 - 1.74iT - 37T^{2} \)
41 \( 1 - 8.09iT - 41T^{2} \)
43 \( 1 - 0.770iT - 43T^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 - 1.03iT - 53T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 - 1.09iT - 67T^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 + 5.96iT - 73T^{2} \)
79 \( 1 - 8.62T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 - 8.16iT - 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.89476800439326094058070956356, −10.83749523356389602737419275308, −9.639367095281897060055677130481, −9.361199892468444654991654407063, −8.332230590295464417747235007464, −6.70378028772140510325561062858, −6.33070967848373999932579008822, −4.87720500921841065151384919377, −3.90927441675529685744337903036, −1.58219246825616697157073117853, 0.17678319186235787895191156627, 2.45531379445463472890451824723, 3.75261485573593655758622436758, 5.84088148432443818301566409165, 6.29519409275639894133831699625, 7.23902417612831251061127717250, 8.095382421159393350405615997642, 9.679437953916926383254515249284, 10.39800936302582037568586425769, 10.77949573523982990776007732215

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