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Properties

Label 1-311-311.234-r0-0-0
Degree $1$
Conductor $311$
Sign $0.665 + 0.746i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
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  + (−0.994 − 0.101i)2-s + (−0.994 + 0.101i)3-s + (0.979 + 0.201i)4-s + (0.688 − 0.724i)5-s + 6-s + (0.347 + 0.937i)7-s + (−0.954 − 0.299i)8-s + (0.979 − 0.201i)9-s + (−0.758 + 0.651i)10-s + (−0.0506 + 0.998i)11-s + (−0.994 − 0.101i)12-s + (0.979 − 0.201i)13-s + (−0.250 − 0.968i)14-s + (−0.612 + 0.790i)15-s + (0.918 + 0.394i)16-s + (−0.758 + 0.651i)17-s + ⋯
L(s)  = 1  + (−0.994 − 0.101i)2-s + (−0.994 + 0.101i)3-s + (0.979 + 0.201i)4-s + (0.688 − 0.724i)5-s + 6-s + (0.347 + 0.937i)7-s + (−0.954 − 0.299i)8-s + (0.979 − 0.201i)9-s + (−0.758 + 0.651i)10-s + (−0.0506 + 0.998i)11-s + (−0.994 − 0.101i)12-s + (0.979 − 0.201i)13-s + (−0.250 − 0.968i)14-s + (−0.612 + 0.790i)15-s + (0.918 + 0.394i)16-s + (−0.758 + 0.651i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.665 + 0.746i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.665 + 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6067229123 + 0.2720591077i\)
\(L(\frac12)\) \(\approx\) \(0.6067229123 + 0.2720591077i\)
\(L(1)\) \(\approx\) \(0.6254838475 + 0.08052779170i\)
\(L(1)\) \(\approx\) \(0.6254838475 + 0.08052779170i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.994 - 0.101i)T \)
3 \( 1 + (-0.994 + 0.101i)T \)
5 \( 1 + (0.688 - 0.724i)T \)
7 \( 1 + (0.347 + 0.937i)T \)
11 \( 1 + (-0.0506 + 0.998i)T \)
13 \( 1 + (0.979 - 0.201i)T \)
17 \( 1 + (-0.758 + 0.651i)T \)
19 \( 1 + (0.688 + 0.724i)T \)
23 \( 1 + (-0.954 - 0.299i)T \)
29 \( 1 + (-0.250 + 0.968i)T \)
31 \( 1 + (-0.758 - 0.651i)T \)
37 \( 1 + (0.347 - 0.937i)T \)
41 \( 1 + (0.528 - 0.848i)T \)
43 \( 1 + (0.347 + 0.937i)T \)
47 \( 1 + (-0.250 + 0.968i)T \)
53 \( 1 + (0.347 + 0.937i)T \)
59 \( 1 + (0.347 + 0.937i)T \)
61 \( 1 + (0.688 + 0.724i)T \)
67 \( 1 + (0.528 + 0.848i)T \)
71 \( 1 + (-0.874 + 0.485i)T \)
73 \( 1 + (0.820 + 0.571i)T \)
79 \( 1 + (0.528 - 0.848i)T \)
83 \( 1 + (-0.612 - 0.790i)T \)
89 \( 1 + (0.347 - 0.937i)T \)
97 \( 1 + (-0.874 + 0.485i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.18366480396160457425140546261, −24.189389888438360640566986729100, −23.58662974722233126374619675295, −22.40606867410270424957062134777, −21.52243086371265199511862486758, −20.640461221554748532686616209816, −19.50162896962085268107185619780, −18.315394312099348257209038594741, −18.05286690416696265111868629921, −17.11599544356409551026713586990, −16.31464700767760042109140424732, −15.511910261079765040528611452302, −14.008216460815395557893079293676, −13.31802961272872603507218415134, −11.44361095258760373875090609189, −11.21650418907014000883739506689, −10.31260214723314580381271120262, −9.41626634116916742425337231967, −8.028502805400572993462297419, −6.94022402085572430371107743099, −6.33459752308246577548852434558, −5.29672509364292918829309080422, −3.5659053630359643971562373380, −1.9731986870819926522242937783, −0.74923216342927251823300639761, 1.32552656333877125132287321726, 2.150622810538121650347011156717, 4.1796326982302805264538270804, 5.63037780313471643913765197918, 6.08449638221670023110024675893, 7.4740965191385037696157995805, 8.66776885193156640167122767509, 9.47969546768862157631135028722, 10.39875104366088180988079645902, 11.338669528531024514422725540504, 12.35088658538970224398737759791, 12.874772641318700847849490002252, 14.70698437338699163142125172491, 15.86240988956094684501400662751, 16.32421097819763039629260946333, 17.57258929776572946578214204623, 17.906883727255867979305367941201, 18.62527754568194559559177708706, 20.10916550058612078331472979673, 20.836487830193460421292986605213, 21.62851350341586763855941701434, 22.519030835513732084036372705903, 23.90710585132096385881179353876, 24.533540489217797462220186719999, 25.40622704547929944651416721319

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