Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.328 + 0.944i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.82i·3-s − 0.284i·5-s − 0.645i·7-s − 0.325·9-s + 1.56i·11-s + 1.12·13-s − 0.519·15-s + (1.35 − 3.89i)17-s − 0.845·19-s − 1.17·21-s + 2.49i·23-s + 4.91·25-s − 4.87i·27-s + 9.28i·29-s − 5.94i·31-s + ⋯
L(s)  = 1  − 1.05i·3-s − 0.127i·5-s − 0.243i·7-s − 0.108·9-s + 0.471i·11-s + 0.311·13-s − 0.134·15-s + (0.328 − 0.944i)17-s − 0.194·19-s − 0.256·21-s + 0.520i·23-s + 0.983·25-s − 0.938i·27-s + 1.72i·29-s − 1.06i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.328 + 0.944i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.328 + 0.944i)$
$L(1)$  $\approx$  $1.932179730$
$L(\frac12)$  $\approx$  $1.932179730$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;17,\;59\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-1.35 + 3.89i)T \)
59 \( 1 - T \)
good3 \( 1 + 1.82iT - 3T^{2} \)
5 \( 1 + 0.284iT - 5T^{2} \)
7 \( 1 + 0.645iT - 7T^{2} \)
11 \( 1 - 1.56iT - 11T^{2} \)
13 \( 1 - 1.12T + 13T^{2} \)
19 \( 1 + 0.845T + 19T^{2} \)
23 \( 1 - 2.49iT - 23T^{2} \)
29 \( 1 - 9.28iT - 29T^{2} \)
31 \( 1 + 5.94iT - 31T^{2} \)
37 \( 1 + 6.09iT - 37T^{2} \)
41 \( 1 + 1.30iT - 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 - 0.319T + 47T^{2} \)
53 \( 1 + 3.07T + 53T^{2} \)
61 \( 1 - 4.55iT - 61T^{2} \)
67 \( 1 - 4.98T + 67T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 - 1.65iT - 79T^{2} \)
83 \( 1 - 4.62T + 83T^{2} \)
89 \( 1 - 1.32T + 89T^{2} \)
97 \( 1 + 6.36iT - 97T^{2} \)
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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

−8.004075275936751887531081044029, −7.35405831774896253886856812560, −6.99350135769725769510219087461, −6.15279881962133913605117724583, −5.29492950015672042423090526281, −4.48837103828015358241101968639, −3.50587692404038614488464107907, −2.48318530663100274397172057898, −1.56100535121083354626224399756, −0.64239784104294278896617014653, 1.12739397734897435808662694310, 2.47214258248942519052754762062, 3.40429028471752051111934343132, 4.09955980250218002276346314963, 4.80948848428683243888171832467, 5.65239024144803044256649208712, 6.35255261475184954828629331275, 7.14442888062386319649938861433, 8.294906986724740881854569534695, 8.550659569592722749045137750584

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