Properties

Degree 1
Conductor 311
Sign $-0.571 + 0.820i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.151 + 0.988i)2-s + (0.151 − 0.988i)3-s + (−0.954 + 0.299i)4-s + (0.347 + 0.937i)5-s + 6-s + (−0.250 − 0.968i)7-s + (−0.440 − 0.897i)8-s + (−0.954 − 0.299i)9-s + (−0.874 + 0.485i)10-s + (0.758 + 0.651i)11-s + (0.151 + 0.988i)12-s + (−0.954 − 0.299i)13-s + (0.918 − 0.394i)14-s + (0.979 − 0.201i)15-s + (0.820 − 0.571i)16-s + (0.874 − 0.485i)17-s + ⋯
L(s,χ)  = 1  + (0.151 + 0.988i)2-s + (0.151 − 0.988i)3-s + (−0.954 + 0.299i)4-s + (0.347 + 0.937i)5-s + 6-s + (−0.250 − 0.968i)7-s + (−0.440 − 0.897i)8-s + (−0.954 − 0.299i)9-s + (−0.874 + 0.485i)10-s + (0.758 + 0.651i)11-s + (0.151 + 0.988i)12-s + (−0.954 − 0.299i)13-s + (0.918 − 0.394i)14-s + (0.979 − 0.201i)15-s + (0.820 − 0.571i)16-s + (0.874 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.571 + 0.820i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.571 + 0.820i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(311\)
\( \varepsilon \)  =  $-0.571 + 0.820i$
motivic weight  =  \(0\)
character  :  $\chi_{311} (116, \cdot )$
Sato-Tate  :  $\mu(62)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 311,\ (1:\ ),\ -0.571 + 0.820i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6350252480 + 1.215275606i$
$L(\frac12,\chi)$  $\approx$  $0.6350252480 + 1.215275606i$
$L(\chi,1)$  $\approx$  0.9446375881 + 0.4037695781i
$L(1,\chi)$  $\approx$  0.9446375881 + 0.4037695781i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.678344934063992266606763965653, −23.75468094897443428183622400143, −22.268990145822548031847666086125, −21.976772761901657208318934753452, −21.16841269316600311428500853032, −20.41718452926123105047615752562, −19.46748083021636489047298227968, −18.79690344909833622364404761039, −17.16855325795850169433750136491, −16.83946243162217799218389768901, −15.43747493912137880564329544280, −14.56921160904232951171439455642, −13.63262468499858377364143642554, −12.44544876648774988718487387256, −11.84215483056918596339760786697, −10.704532698534986188545864721139, −9.60128389510756227584320926543, −9.077507019138382911845256835649, −8.36700740477240064554123041883, −6.02302421230632881360570832244, −5.15295279585885045473365153440, −4.32022085889708259262024722059, −3.11768005224015532692933249356, −2.059092634098319288358862949292, −0.41184563506904581442230624808, 1.206605303911243154664003864163, 2.87668209433686295206588135297, 3.99759007071166991772874265927, 5.56611962916022165207167787749, 6.51520110708220259484050898001, 7.351228379794577251531597576389, 7.72537268106445845450786757944, 9.38008955395565536066355286355, 10.13411718096685137218757330283, 11.70023069453312769866012992486, 12.73965860262893637309667913078, 13.61334992921203191742948673468, 14.503058233988130643083818769877, 14.81511721081476623998347064042, 16.46733222448577771459707912736, 17.35400637459172852077126425362, 17.79456146131185050847569709759, 18.9716337505401210881636149170, 19.54583725518563028914029074440, 20.90915176284446549319063602992, 22.25204400386952134730744571209, 23.014818261590459697712653351028, 23.34219125696081418899848019780, 24.70430534431889975557462768238, 25.18951819517932386763284346015

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