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Properties

Label 2-3311-3311.706-c0-0-1
Degree $2$
Conductor $3311$
Sign $-0.934 + 0.354i$
Analytic cond. $1.65240$
Root an. cond. $1.28545$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 − 1.36i)2-s + (−0.115 − 2.57i)4-s + (0.913 − 0.406i)7-s + (−2.25 − 1.97i)8-s + (−0.998 + 0.0598i)9-s + (0.995 − 0.0896i)11-s + (0.638 − 1.78i)14-s + (−3.07 + 0.276i)16-s + (−1.22 + 1.44i)18-s + (1.18 − 1.47i)22-s + (1.96 + 0.295i)23-s + (−0.525 − 0.850i)25-s + (−1.15 − 2.30i)28-s + (−1.54 − 0.0462i)29-s + (−1.77 + 2.22i)32-s + ⋯
L(s)  = 1  + (1.30 − 1.36i)2-s + (−0.115 − 2.57i)4-s + (0.913 − 0.406i)7-s + (−2.25 − 1.97i)8-s + (−0.998 + 0.0598i)9-s + (0.995 − 0.0896i)11-s + (0.638 − 1.78i)14-s + (−3.07 + 0.276i)16-s + (−1.22 + 1.44i)18-s + (1.18 − 1.47i)22-s + (1.96 + 0.295i)23-s + (−0.525 − 0.850i)25-s + (−1.15 − 2.30i)28-s + (−1.54 − 0.0462i)29-s + (−1.77 + 2.22i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.934 + 0.354i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (706, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.934 + 0.354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.581424297\)
\(L(\frac12)\) \(\approx\) \(2.581424297\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.913 + 0.406i)T \)
11 \( 1 + (-0.995 + 0.0896i)T \)
43 \( 1 + (-0.447 - 0.894i)T \)
good2 \( 1 + (-1.30 + 1.36i)T + (-0.0448 - 0.998i)T^{2} \)
3 \( 1 + (0.998 - 0.0598i)T^{2} \)
5 \( 1 + (0.525 + 0.850i)T^{2} \)
13 \( 1 + (-0.337 - 0.941i)T^{2} \)
17 \( 1 + (-0.646 - 0.762i)T^{2} \)
19 \( 1 + (0.873 - 0.486i)T^{2} \)
23 \( 1 + (-1.96 - 0.295i)T + (0.955 + 0.294i)T^{2} \)
29 \( 1 + (1.54 + 0.0462i)T + (0.998 + 0.0598i)T^{2} \)
31 \( 1 + (-0.887 - 0.460i)T^{2} \)
37 \( 1 + (1.82 + 0.192i)T + (0.978 + 0.207i)T^{2} \)
41 \( 1 + (0.936 - 0.351i)T^{2} \)
47 \( 1 + (-0.858 - 0.512i)T^{2} \)
53 \( 1 + (0.0297 - 1.99i)T + (-0.999 - 0.0299i)T^{2} \)
59 \( 1 + (-0.134 + 0.990i)T^{2} \)
61 \( 1 + (-0.887 + 0.460i)T^{2} \)
67 \( 1 + (0.676 - 1.72i)T + (-0.733 - 0.680i)T^{2} \)
71 \( 1 + (-0.371 + 0.186i)T + (0.599 - 0.800i)T^{2} \)
73 \( 1 + (0.280 - 0.959i)T^{2} \)
79 \( 1 + (-1.32 + 1.19i)T + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (-0.842 - 0.538i)T^{2} \)
89 \( 1 + (0.365 + 0.930i)T^{2} \)
97 \( 1 + (0.393 - 0.919i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.937939929934760540019945614824, −7.67323870053750789319067362464, −6.70688165717442181311164625696, −5.81285197962553305683740085668, −5.23183796638669647202460306229, −4.46770044887224554670635140301, −3.73045605803577862597530243660, −2.97277966733489136797881106405, −1.97807175640256800734856808589, −1.08204371802755680847856303075, 1.99509864273141402457399558423, 3.25089075622017782369521724982, 3.79064082655898348772602680279, 4.94914937516872372375573341629, 5.27307029164115405652351342867, 5.99630851196126853932899539942, 6.86987057081002403245829142536, 7.35345007578659930281342453294, 8.273127005975254146160451696185, 8.801823018419995293208465360835

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