Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 59 $
Sign $-0.964 + 0.264i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (0.869 − 1.49i)3-s + 4-s − 1.66i·5-s + (−0.869 + 1.49i)6-s − 3.57·7-s − 8-s + (−1.48 − 2.60i)9-s + 1.66i·10-s − 4.82·11-s + (0.869 − 1.49i)12-s + 0.418i·13-s + 3.57·14-s + (−2.48 − 1.44i)15-s + 16-s + 0.742i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.502 − 0.864i)3-s + 0.5·4-s − 0.742i·5-s + (−0.355 + 0.611i)6-s − 1.35·7-s − 0.353·8-s + (−0.495 − 0.868i)9-s + 0.524i·10-s − 1.45·11-s + (0.251 − 0.432i)12-s + 0.116i·13-s + 0.956·14-s + (−0.642 − 0.372i)15-s + 0.250·16-s + 0.180i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(354\)    =    \(2 \cdot 3 \cdot 59\)
\( \varepsilon \)  =  $-0.964 + 0.264i$
motivic weight  =  \(1\)
character  :  $\chi_{354} (353, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 354,\ (\ :1/2),\ -0.964 + 0.264i)$
$L(1)$  $\approx$  $0.0779952 - 0.579371i$
$L(\frac12)$  $\approx$  $0.0779952 - 0.579371i$
$L(\frac{3}{2})$   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 \( 1 + (-0.869 + 1.49i)T \)
59 \( 1 + (-5.47 - 5.38i)T \)
good5 \( 1 + 1.66iT - 5T^{2} \)
7 \( 1 + 3.57T + 7T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 - 0.418iT - 13T^{2} \)
17 \( 1 - 0.742iT - 17T^{2} \)
19 \( 1 - 1.24T + 19T^{2} \)
23 \( 1 - 1.25T + 23T^{2} \)
29 \( 1 - 4.78iT - 29T^{2} \)
31 \( 1 + 10.4iT - 31T^{2} \)
37 \( 1 + 6.02iT - 37T^{2} \)
41 \( 1 + 9.24iT - 41T^{2} \)
43 \( 1 - 3.58iT - 43T^{2} \)
47 \( 1 + 8.39T + 47T^{2} \)
53 \( 1 + 9.66iT - 53T^{2} \)
61 \( 1 + 9.08iT - 61T^{2} \)
67 \( 1 + 13.0iT - 67T^{2} \)
71 \( 1 - 4.01iT - 71T^{2} \)
73 \( 1 - 10.6iT - 73T^{2} \)
79 \( 1 - 3.55T + 79T^{2} \)
83 \( 1 + 2.87T + 83T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 - 11.5iT - 97T^{2} \)
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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

−10.95581581726219086221310924872, −9.831905273571093651721480532359, −9.151214928524464038968547652122, −8.252311428157221318248102208910, −7.42237729760466416561537149257, −6.47320942826670643312044386056, −5.38803806778467572010911440168, −3.43316960280639529696700690842, −2.28211974081840585064491615231, −0.44038283552706877865809875496, 2.75059940896852471278931911046, 3.23803952970376798417246378690, 4.99580644492729200764942043894, 6.27860994755856958010651042599, 7.31573079607611625570094090941, 8.296828760314077096337823107205, 9.280679739517931336114181371808, 10.25663725548389229507880189388, 10.37770811813735534936973403293, 11.54385675600396742251036226615

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