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Properties

Label 1-316-316.131-r1-0-0
Degree $1$
Conductor $316$
Sign $-0.327 + 0.944i$
Analytic cond. $33.9589$
Root an. cond. $33.9589$
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  + (0.970 + 0.239i)3-s + (−0.748 + 0.663i)5-s + (0.970 + 0.239i)7-s + (0.885 + 0.464i)9-s + (0.748 + 0.663i)11-s + (−0.354 + 0.935i)13-s + (−0.885 + 0.464i)15-s + (−0.354 + 0.935i)17-s + (−0.120 − 0.992i)19-s + (0.885 + 0.464i)21-s − 23-s + (0.120 − 0.992i)25-s + (0.748 + 0.663i)27-s + (0.885 − 0.464i)29-s + (−0.568 − 0.822i)31-s + ⋯
L(s)  = 1  + (0.970 + 0.239i)3-s + (−0.748 + 0.663i)5-s + (0.970 + 0.239i)7-s + (0.885 + 0.464i)9-s + (0.748 + 0.663i)11-s + (−0.354 + 0.935i)13-s + (−0.885 + 0.464i)15-s + (−0.354 + 0.935i)17-s + (−0.120 − 0.992i)19-s + (0.885 + 0.464i)21-s − 23-s + (0.120 − 0.992i)25-s + (0.748 + 0.663i)27-s + (0.885 − 0.464i)29-s + (−0.568 − 0.822i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 316 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 316 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(316\)    =    \(2^{2} \cdot 79\)
Sign: $-0.327 + 0.944i$
Analytic conductor: \(33.9589\)
Root analytic conductor: \(33.9589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{316} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 316,\ (1:\ ),\ -0.327 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.470477961 + 2.065195821i\)
\(L(\frac12)\) \(\approx\) \(1.470477961 + 2.065195821i\)
\(L(1)\) \(\approx\) \(1.344740882 + 0.5896751535i\)
\(L(1)\) \(\approx\) \(1.344740882 + 0.5896751535i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
79 \( 1 \)
good3 \( 1 + (0.970 + 0.239i)T \)
5 \( 1 + (-0.748 + 0.663i)T \)
7 \( 1 + (0.970 + 0.239i)T \)
11 \( 1 + (0.748 + 0.663i)T \)
13 \( 1 + (-0.354 + 0.935i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (-0.120 - 0.992i)T \)
23 \( 1 - T \)
29 \( 1 + (0.885 - 0.464i)T \)
31 \( 1 + (-0.568 - 0.822i)T \)
37 \( 1 + (0.120 + 0.992i)T \)
41 \( 1 + (-0.748 + 0.663i)T \)
43 \( 1 + (0.748 - 0.663i)T \)
47 \( 1 + (-0.120 + 0.992i)T \)
53 \( 1 + (-0.970 + 0.239i)T \)
59 \( 1 + (0.354 + 0.935i)T \)
61 \( 1 + (0.120 + 0.992i)T \)
67 \( 1 + (-0.568 + 0.822i)T \)
71 \( 1 + (0.970 - 0.239i)T \)
73 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (0.354 - 0.935i)T \)
89 \( 1 + (-0.970 - 0.239i)T \)
97 \( 1 + (0.120 + 0.992i)T \)
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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

−24.72514407453014701695813527722, −24.05310105854979391718556079284, −23.13367954213987223734747122605, −21.85427393786407045597747566329, −20.83794812949445117319116823843, −20.13022005213189634540054154735, −19.606456836335446991937220429238, −18.47554879687970833771469431920, −17.58822904146881874208092480372, −16.36567867395862696386828850954, −15.56379246189628650955708322129, −14.44617134643437634982069486707, −13.96128720735910866295435250933, −12.65914731920870878035147668505, −11.95198026907651067268377878729, −10.80787360478056394244353574751, −9.50844243235185763348405791190, −8.432499930664854003999928574169, −7.99677746376419494348216992202, −6.9585040989241944900498091384, −5.3160273010083856269007818043, −4.19952208185229803127721602342, −3.31280583246372337010743341110, −1.796591831452358865928950850781, −0.663006681951210628642943368855, 1.68209004056683606622512989795, 2.63092838687666764672725875038, 4.109481141648202767611725160758, 4.51414827318509008297357463170, 6.47981701168576066217028553291, 7.452589009775090299064667762431, 8.29224288956386228940955988785, 9.21510655754132571727916972517, 10.32856972791947466285899369077, 11.40489057331350762321455521620, 12.18820888035107198392049221257, 13.60552565760480885387670259415, 14.556694861632921169279094668378, 14.984120493816132025547263176726, 15.83265946021075358301122487929, 17.1741248151461547749840192494, 18.173545306966114719874117647252, 19.18947193005050312736097436802, 19.76238605909590577355515244533, 20.65715272709348267646693211263, 21.77389242064055051739464443256, 22.234770750742825087737666501232, 23.86026443850268632753417804348, 24.12738717552827393887316446370, 25.42417526577564195070664143086

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