Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 67
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.26e10·2-s − 1.28e15·3-s + 3.67e20·4-s − 1.61e23·5-s − 2.91e25·6-s + 2.93e28·7-s + 4.99e30·8-s − 9.10e31·9-s − 3.65e33·10-s + 1.03e35·11-s − 4.71e35·12-s − 9.30e35·13-s + 6.66e38·14-s + 2.07e38·15-s + 5.90e40·16-s + 2.52e41·17-s − 2.06e42·18-s − 4.10e41·19-s − 5.92e43·20-s − 3.77e43·21-s + 2.36e45·22-s − 7.28e44·23-s − 6.40e45·24-s − 4.17e46·25-s − 2.11e46·26-s + 2.36e47·27-s + 1.07e49·28-s + ⋯
L(s)  = 1  + 1.86·2-s − 0.133·3-s + 2.49·4-s − 0.619·5-s − 0.249·6-s + 1.43·7-s + 2.78·8-s − 0.982·9-s − 1.15·10-s + 1.34·11-s − 0.332·12-s − 0.0448·13-s + 2.68·14-s + 0.0825·15-s + 2.71·16-s + 1.51·17-s − 1.83·18-s − 0.0596·19-s − 1.54·20-s − 0.191·21-s + 2.52·22-s − 0.175·23-s − 0.371·24-s − 0.616·25-s − 0.0837·26-s + 0.264·27-s + 3.57·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(68-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+67/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(67\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1,\ (\ :67/2),\ 1)\)
\(L(34)\)  \(\approx\)  \(6.502194688\)
\(L(\frac12)\)  \(\approx\)  \(6.502194688\)
\(L(\frac{69}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 - 2.26e10T + 1.47e20T^{2} \)
3 \( 1 + 1.28e15T + 9.27e31T^{2} \)
5 \( 1 + 1.61e23T + 6.77e46T^{2} \)
7 \( 1 - 2.93e28T + 4.18e56T^{2} \)
11 \( 1 - 1.03e35T + 5.93e69T^{2} \)
13 \( 1 + 9.30e35T + 4.30e74T^{2} \)
17 \( 1 - 2.52e41T + 2.75e82T^{2} \)
19 \( 1 + 4.10e41T + 4.74e85T^{2} \)
23 \( 1 + 7.28e44T + 1.72e91T^{2} \)
29 \( 1 + 8.66e48T + 9.56e97T^{2} \)
31 \( 1 - 5.42e49T + 8.34e99T^{2} \)
37 \( 1 - 2.44e52T + 1.17e105T^{2} \)
41 \( 1 - 8.84e52T + 1.13e108T^{2} \)
43 \( 1 + 3.35e54T + 2.76e109T^{2} \)
47 \( 1 + 1.34e56T + 1.07e112T^{2} \)
53 \( 1 + 3.87e57T + 3.36e115T^{2} \)
59 \( 1 + 2.62e59T + 4.43e118T^{2} \)
61 \( 1 + 4.77e59T + 4.14e119T^{2} \)
67 \( 1 + 2.36e61T + 2.22e122T^{2} \)
71 \( 1 - 2.36e61T + 1.08e124T^{2} \)
73 \( 1 - 7.86e61T + 6.96e124T^{2} \)
79 \( 1 + 2.74e63T + 1.38e127T^{2} \)
83 \( 1 - 9.16e63T + 3.78e128T^{2} \)
89 \( 1 + 3.36e65T + 4.06e130T^{2} \)
97 \( 1 - 6.50e66T + 1.29e133T^{2} \)
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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

−16.84400840255076415352011629698, −14.86878587814042581692458752428, −14.15088345273854872768568121710, −11.96805756271756216134880451392, −11.35849997136106132905499674166, −7.79914147839544650450516891186, −5.94469613750623119616267567635, −4.63299702481245344620824972548, −3.37379077599699753982702523477, −1.57226775393031713484103606981, 1.57226775393031713484103606981, 3.37379077599699753982702523477, 4.63299702481245344620824972548, 5.94469613750623119616267567635, 7.79914147839544650450516891186, 11.35849997136106132905499674166, 11.96805756271756216134880451392, 14.15088345273854872768568121710, 14.86878587814042581692458752428, 16.84400840255076415352011629698

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