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Properties

Label 1-371-371.108-r0-0-0
Degree $1$
Conductor $371$
Sign $0.326 - 0.945i$
Analytic cond. $1.72291$
Root an. cond. $1.72291$
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  + (−0.600 + 0.799i)2-s + (−0.999 + 0.0402i)3-s + (−0.278 − 0.960i)4-s + (−0.0804 − 0.996i)5-s + (0.568 − 0.822i)6-s + (0.935 + 0.354i)8-s + (0.996 − 0.0804i)9-s + (0.845 + 0.534i)10-s + (0.200 − 0.979i)11-s + (0.316 + 0.948i)12-s + (0.970 − 0.239i)13-s + (0.120 + 0.992i)15-s + (−0.845 + 0.534i)16-s + (−0.632 + 0.774i)17-s + (−0.534 + 0.845i)18-s + (−0.960 − 0.278i)19-s + ⋯
L(s)  = 1  + (−0.600 + 0.799i)2-s + (−0.999 + 0.0402i)3-s + (−0.278 − 0.960i)4-s + (−0.0804 − 0.996i)5-s + (0.568 − 0.822i)6-s + (0.935 + 0.354i)8-s + (0.996 − 0.0804i)9-s + (0.845 + 0.534i)10-s + (0.200 − 0.979i)11-s + (0.316 + 0.948i)12-s + (0.970 − 0.239i)13-s + (0.120 + 0.992i)15-s + (−0.845 + 0.534i)16-s + (−0.632 + 0.774i)17-s + (−0.534 + 0.845i)18-s + (−0.960 − 0.278i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(371\)    =    \(7 \cdot 53\)
Sign: $0.326 - 0.945i$
Analytic conductor: \(1.72291\)
Root analytic conductor: \(1.72291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{371} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 371,\ (0:\ ),\ 0.326 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4324680168 - 0.3081334796i\)
\(L(\frac12)\) \(\approx\) \(0.4324680168 - 0.3081334796i\)
\(L(1)\) \(\approx\) \(0.5659867243 + 0.02596684985i\)
\(L(1)\) \(\approx\) \(0.5659867243 + 0.02596684985i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.600 + 0.799i)T \)
3 \( 1 + (-0.999 + 0.0402i)T \)
5 \( 1 + (-0.0804 - 0.996i)T \)
11 \( 1 + (0.200 - 0.979i)T \)
13 \( 1 + (0.970 - 0.239i)T \)
17 \( 1 + (-0.632 + 0.774i)T \)
19 \( 1 + (-0.960 - 0.278i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.748 - 0.663i)T \)
31 \( 1 + (0.316 - 0.948i)T \)
37 \( 1 + (0.845 - 0.534i)T \)
41 \( 1 + (-0.663 + 0.748i)T \)
43 \( 1 + (-0.885 + 0.464i)T \)
47 \( 1 + (-0.428 - 0.903i)T \)
59 \( 1 + (-0.996 - 0.0804i)T \)
61 \( 1 + (0.774 - 0.632i)T \)
67 \( 1 + (0.960 - 0.278i)T \)
71 \( 1 + (-0.464 - 0.885i)T \)
73 \( 1 + (0.774 + 0.632i)T \)
79 \( 1 + (0.391 - 0.919i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.987 - 0.160i)T \)
97 \( 1 + (-0.568 - 0.822i)T \)
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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

−25.11776578167070571306681825379, −23.4510328360443335392138369871, −22.937071822653304311589873025921, −22.16904764090318623544222209662, −21.37116169502482158267729529924, −20.454855110703140530598949824972, −19.30872888735140500219751374499, −18.4826337241476699342146544196, −17.9565994145708255161620871923, −17.157786217190811773124918018243, −16.14603374927877238364004816569, −15.19364541884932777966221333538, −13.83050007554788922771309952410, −12.782032065953985437298711977738, −11.92882214217481985184628925765, −11.054253305452806344340050221662, −10.5257307722493564003103894199, −9.59620464979760983903938194197, −8.366063475990620929167496337930, −6.969740395455335859354951388479, −6.61433677463538079258461338648, −4.85730774973940839435177393679, −3.89678303927109929773682264161, −2.56993161089257894328442863944, −1.35653796909655415617203500982, 0.510670720054326638872261995994, 1.56885979532178234066147356534, 4.023286776790738384077685554162, 4.958425691972625014184475146063, 5.99007993820773132268875313285, 6.51056917139987558709240423370, 8.026813235364603808225589768282, 8.685749747476076235509902805, 9.72826813649234969492272045861, 10.87596169903349698387066913410, 11.49543553482783078904700898976, 13.00297992108959776897039281513, 13.47732538990451585130597770970, 15.13522984756349662358343907367, 15.77494911775763967695177643786, 16.71551274630237650154833790314, 17.09540270083624469999820649161, 18.03832982291413133940333078313, 18.99643184525818691399326666270, 19.806171256035201572501381385754, 21.10268042682953988805001434216, 21.875556410903476993008305465816, 23.30994985830872615688131266834, 23.47515683965173674829308705935, 24.52278686920287339509857728265

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