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Properties

Label 1-311-311.296-r1-0-0
Degree $1$
Conductor $311$
Sign $-0.606 - 0.795i$
Analytic cond. $33.4215$
Root an. cond. $33.4215$
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  + (−0.954 − 0.299i)2-s + (−0.954 + 0.299i)3-s + (0.820 + 0.571i)4-s + (−0.758 − 0.651i)5-s + 6-s + (−0.874 − 0.485i)7-s + (−0.612 − 0.790i)8-s + (0.820 − 0.571i)9-s + (0.528 + 0.848i)10-s + (−0.151 + 0.988i)11-s + (−0.954 − 0.299i)12-s + (0.820 − 0.571i)13-s + (0.688 + 0.724i)14-s + (0.918 + 0.394i)15-s + (0.347 + 0.937i)16-s + (−0.528 − 0.848i)17-s + ⋯
L(s)  = 1  + (−0.954 − 0.299i)2-s + (−0.954 + 0.299i)3-s + (0.820 + 0.571i)4-s + (−0.758 − 0.651i)5-s + 6-s + (−0.874 − 0.485i)7-s + (−0.612 − 0.790i)8-s + (0.820 − 0.571i)9-s + (0.528 + 0.848i)10-s + (−0.151 + 0.988i)11-s + (−0.954 − 0.299i)12-s + (0.820 − 0.571i)13-s + (0.688 + 0.724i)14-s + (0.918 + 0.394i)15-s + (0.347 + 0.937i)16-s + (−0.528 − 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.606 - 0.795i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ -0.606 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1714132067 - 0.3463504552i\)
\(L(\frac12)\) \(\approx\) \(0.1714132067 - 0.3463504552i\)
\(L(1)\) \(\approx\) \(0.4132084702 - 0.1034908339i\)
\(L(1)\) \(\approx\) \(0.4132084702 - 0.1034908339i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.954 - 0.299i)T \)
3 \( 1 + (-0.954 + 0.299i)T \)
5 \( 1 + (-0.758 - 0.651i)T \)
7 \( 1 + (-0.874 - 0.485i)T \)
11 \( 1 + (-0.151 + 0.988i)T \)
13 \( 1 + (0.820 - 0.571i)T \)
17 \( 1 + (-0.528 - 0.848i)T \)
19 \( 1 + (0.758 - 0.651i)T \)
23 \( 1 + (0.612 + 0.790i)T \)
29 \( 1 + (-0.688 + 0.724i)T \)
31 \( 1 + (-0.528 + 0.848i)T \)
37 \( 1 + (0.874 - 0.485i)T \)
41 \( 1 + (0.994 + 0.101i)T \)
43 \( 1 + (0.874 + 0.485i)T \)
47 \( 1 + (0.688 - 0.724i)T \)
53 \( 1 + (-0.874 - 0.485i)T \)
59 \( 1 + (0.874 + 0.485i)T \)
61 \( 1 + (0.758 - 0.651i)T \)
67 \( 1 + (-0.994 + 0.101i)T \)
71 \( 1 + (0.0506 - 0.998i)T \)
73 \( 1 + (-0.250 + 0.968i)T \)
79 \( 1 + (-0.994 - 0.101i)T \)
83 \( 1 + (0.918 - 0.394i)T \)
89 \( 1 + (-0.874 + 0.485i)T \)
97 \( 1 + (0.0506 - 0.998i)T \)
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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

−25.44656989999947356542405476456, −24.31373988272568649803147718385, −23.75112259262925910261611687306, −22.749100622819712387827518744116, −22.01594328715890632827445379885, −20.75013091780063074027135199444, −19.35927925379338006869857883657, −18.800095392179036613937761106620, −18.42633075242402357725839515714, −17.14286501139143453691518156771, −16.180253954346625095108953212094, −15.858107095620966361365267349601, −14.69638600198447481292381436800, −13.24773613836857149093688891803, −12.06099711225947638806994904969, −11.18326212701151657170908149347, −10.674542625115862183249622548343, −9.42382655135846447591474619474, −8.28242507850255468791083047891, −7.297451072841415569405832364917, −6.23861187720922781682206645393, −5.87112878034377179107920079570, −3.92385975260159167762231502982, −2.50506600486463483491264716517, −0.873028989268912590299184005762, 0.27852711435939422442588696898, 1.1889315124519283425387859323, 3.17532124220374282851038683536, 4.23219149468859002695082961475, 5.519888259332949944180836370673, 6.98203683635694337268866465893, 7.45575774114904385636589839958, 9.06182082449756977348752430705, 9.65481037098545496575000339736, 10.8384230549621013328077135754, 11.44043225410318642141849625549, 12.571737163823900310450562180074, 13.07279499450020040247163377441, 15.30825399328722084300644757309, 15.97767323174415897253563229951, 16.43868133361082017164593484320, 17.58531586139576313611624257463, 18.137969306911378858779994826664, 19.37496200079123452075258908341, 20.22293794619588139731248330707, 20.75309004238098356182574094953, 22.084533935743456072156561829480, 22.96444780443214071100868730774, 23.6702908034825018825467952962, 24.82460236904407142816169039663

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