Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 59 $
Sign $-0.247 + 0.968i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (0.0738 − 1.73i)3-s + 4-s − 2.30i·5-s + (−0.0738 + 1.73i)6-s + 3.25·7-s − 8-s + (−2.98 − 0.255i)9-s + 2.30i·10-s + 4.56·11-s + (0.0738 − 1.73i)12-s − 2.44i·13-s − 3.25·14-s + (−3.98 − 0.170i)15-s + 16-s + 2.52i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.0426 − 0.999i)3-s + 0.5·4-s − 1.03i·5-s + (−0.0301 + 0.706i)6-s + 1.22·7-s − 0.353·8-s + (−0.996 − 0.0851i)9-s + 0.728i·10-s + 1.37·11-s + (0.0213 − 0.499i)12-s − 0.677i·13-s − 0.869·14-s + (−1.02 − 0.0439i)15-s + 0.250·16-s + 0.613i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(354\)    =    \(2 \cdot 3 \cdot 59\)
\( \varepsilon \)  =  $-0.247 + 0.968i$
motivic weight  =  \(1\)
character  :  $\chi_{354} (353, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 354,\ (\ :1/2),\ -0.247 + 0.968i)$
$L(1)$  $\approx$  $0.671038 - 0.863923i$
$L(\frac12)$  $\approx$  $0.671038 - 0.863923i$
$L(\frac{3}{2})$   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 \( 1 + (-0.0738 + 1.73i)T \)
59 \( 1 + (-7.51 - 1.58i)T \)
good5 \( 1 + 2.30iT - 5T^{2} \)
7 \( 1 - 3.25T + 7T^{2} \)
11 \( 1 - 4.56T + 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 - 2.52iT - 17T^{2} \)
19 \( 1 + 1.31T + 19T^{2} \)
23 \( 1 + 8.45T + 23T^{2} \)
29 \( 1 + 7.58iT - 29T^{2} \)
31 \( 1 - 1.81iT - 31T^{2} \)
37 \( 1 - 2.83iT - 37T^{2} \)
41 \( 1 - 5.77iT - 41T^{2} \)
43 \( 1 + 11.5iT - 43T^{2} \)
47 \( 1 - 0.674T + 47T^{2} \)
53 \( 1 - 8.22iT - 53T^{2} \)
61 \( 1 - 0.133iT - 61T^{2} \)
67 \( 1 + 3.33iT - 67T^{2} \)
71 \( 1 - 6.35iT - 71T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 9.52T + 83T^{2} \)
89 \( 1 - 4.28T + 89T^{2} \)
97 \( 1 - 8.58iT - 97T^{2} \)
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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

−11.48979797641064989346758236983, −10.23677226276624467466012370570, −8.989955220328544475247476141497, −8.317904832718459719458845950778, −7.79590360237408163983920673065, −6.49640449212890766709951904877, −5.54151165007494036233098631018, −4.11011229424872697636413721934, −2.02671837181188182544339155016, −1.04306865061571909889204037014, 2.00777769652594529976108254401, 3.54850221766906354572284753009, 4.64291705430241087043772374442, 6.07298102107745027774432939384, 7.06215379438428731673182071306, 8.201073368424374997376637712734, 9.070450363969016997405637630344, 9.885065356662206620869364684888, 10.84452985732150396477765132306, 11.38345012159794717105581159009

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