Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.0805 - 0.996i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.24i·3-s + 0.276i·5-s − 5.07i·7-s − 7.50·9-s − 3.78i·11-s + 3.39·13-s + 0.896·15-s + (0.332 + 4.10i)17-s − 7.59·19-s − 16.4·21-s − 8.61i·23-s + 4.92·25-s + 14.6i·27-s + 8.08i·29-s + 6.90i·31-s + ⋯
L(s)  = 1  − 1.87i·3-s + 0.123i·5-s − 1.91i·7-s − 2.50·9-s − 1.14i·11-s + 0.942·13-s + 0.231·15-s + (0.0805 + 0.996i)17-s − 1.74·19-s − 3.58·21-s − 1.79i·23-s + 0.984·25-s + 2.81i·27-s + 1.50i·29-s + 1.24i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0805 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.0805 - 0.996i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.0805 - 0.996i)$
$L(1)$  $\approx$  $1.074922411$
$L(\frac12)$  $\approx$  $1.074922411$
$L(\frac{3}{2})$   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 \( 1 + (-0.332 - 4.10i)T \)
59 \( 1 - T \)
good3 \( 1 + 3.24iT - 3T^{2} \)
5 \( 1 - 0.276iT - 5T^{2} \)
7 \( 1 + 5.07iT - 7T^{2} \)
11 \( 1 + 3.78iT - 11T^{2} \)
13 \( 1 - 3.39T + 13T^{2} \)
19 \( 1 + 7.59T + 19T^{2} \)
23 \( 1 + 8.61iT - 23T^{2} \)
29 \( 1 - 8.08iT - 29T^{2} \)
31 \( 1 - 6.90iT - 31T^{2} \)
37 \( 1 + 5.47iT - 37T^{2} \)
41 \( 1 + 5.51iT - 41T^{2} \)
43 \( 1 + 4.87T + 43T^{2} \)
47 \( 1 + 5.79T + 47T^{2} \)
53 \( 1 + 3.84T + 53T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 - 8.67T + 67T^{2} \)
71 \( 1 + 0.288iT - 71T^{2} \)
73 \( 1 + 1.40iT - 73T^{2} \)
79 \( 1 - 11.0iT - 79T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 - 3.59T + 89T^{2} \)
97 \( 1 + 0.104iT - 97T^{2} \)
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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

−7.995436463534311589396359741628, −6.83245078946661826480529654044, −6.71106051503118243440310600666, −6.18312792999556536894909327058, −4.98309428582679629863438758647, −3.81077474901099595955970663063, −3.19229454933149424319408736627, −1.92141711508200708663675921444, −1.09265779242630714495834290690, −0.33013959213271772317237402432, 2.06195110421515693327517631834, 2.82586518356345647621004120638, 3.70439912636255432756455250709, 4.64737363458540344827154389495, 5.02477649633778901431484487199, 5.89792364751574637437851302909, 6.37290056188270643141437170029, 7.87867792131632054929308056071, 8.577146915335156140902600507273, 9.072964039003629759922970028912

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