Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 59 $
Sign $0.883 + 0.468i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (1.64 + 0.548i)3-s + 4-s − 2.54i·5-s + (−1.64 − 0.548i)6-s − 0.0900·7-s − 8-s + (2.39 + 1.80i)9-s + 2.54i·10-s + 2.40·11-s + (1.64 + 0.548i)12-s − 2.81i·13-s + 0.0900·14-s + (1.39 − 4.18i)15-s + 16-s − 3.17i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.948 + 0.316i)3-s + 0.5·4-s − 1.13i·5-s + (−0.670 − 0.223i)6-s − 0.0340·7-s − 0.353·8-s + (0.799 + 0.600i)9-s + 0.805i·10-s + 0.725·11-s + (0.474 + 0.158i)12-s − 0.781i·13-s + 0.0240·14-s + (0.360 − 1.08i)15-s + 0.250·16-s − 0.769i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(354\)    =    \(2 \cdot 3 \cdot 59\)
\( \varepsilon \)  =  $0.883 + 0.468i$
motivic weight  =  \(1\)
character  :  $\chi_{354} (353, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 354,\ (\ :1/2),\ 0.883 + 0.468i)$
$L(1)$  $\approx$  $1.33372 - 0.331505i$
$L(\frac12)$  $\approx$  $1.33372 - 0.331505i$
$L(\frac{3}{2})$   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 \( 1 + (-1.64 - 0.548i)T \)
59 \( 1 + (7.57 + 1.26i)T \)
good5 \( 1 + 2.54iT - 5T^{2} \)
7 \( 1 + 0.0900T + 7T^{2} \)
11 \( 1 - 2.40T + 11T^{2} \)
13 \( 1 + 2.81iT - 13T^{2} \)
17 \( 1 + 3.17iT - 17T^{2} \)
19 \( 1 + 2.49T + 19T^{2} \)
23 \( 1 - 3.09T + 23T^{2} \)
29 \( 1 - 6.21iT - 29T^{2} \)
31 \( 1 - 2.16iT - 31T^{2} \)
37 \( 1 + 11.5iT - 37T^{2} \)
41 \( 1 - 0.762iT - 41T^{2} \)
43 \( 1 - 7.62iT - 43T^{2} \)
47 \( 1 - 7.90T + 47T^{2} \)
53 \( 1 - 3.58iT - 53T^{2} \)
61 \( 1 - 3.56iT - 61T^{2} \)
67 \( 1 - 14.5iT - 67T^{2} \)
71 \( 1 + 4.84iT - 71T^{2} \)
73 \( 1 - 6.03iT - 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + 1.68T + 83T^{2} \)
89 \( 1 + 8.62T + 89T^{2} \)
97 \( 1 - 0.394iT - 97T^{2} \)
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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

−11.18113565416292128179456259927, −10.24328521043390961300177006129, −9.126592138082149217765779389034, −8.936383291407623676829683691102, −7.915149812384793150157477092733, −6.97148909469130670501199459708, −5.39947563441634830696600012268, −4.26681078689955658898526289790, −2.88601089769149407945214798624, −1.27720154074517642947142155742, 1.78897408888218298073846662620, 2.97943538705507058398495672884, 4.12301538351805922609017673628, 6.36207272333723773084184953531, 6.82423544294330783726864959602, 7.85830535104282174396499032323, 8.756711246648668336405211795235, 9.608454956458989582128727098797, 10.45126332055325435296187548133, 11.39093671536948255320283623828

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