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Properties

Label 1-31e2-961.76-r0-0-0
Degree $1$
Conductor $961$
Sign $-0.849 + 0.527i$
Analytic cond. $4.46286$
Root an. cond. $4.46286$
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  + (0.960 − 0.279i)2-s + (−0.899 − 0.437i)3-s + (0.843 − 0.537i)4-s + (−0.315 − 0.948i)5-s + (−0.985 − 0.168i)6-s + (−0.824 + 0.565i)7-s + (0.659 − 0.752i)8-s + (0.617 + 0.786i)9-s + (−0.568 − 0.822i)10-s + (0.124 − 0.992i)11-s + (−0.993 + 0.114i)12-s + (−0.767 + 0.641i)13-s + (−0.633 + 0.773i)14-s + (−0.131 + 0.991i)15-s + (0.422 − 0.906i)16-s + (0.708 − 0.705i)17-s + ⋯
L(s)  = 1  + (0.960 − 0.279i)2-s + (−0.899 − 0.437i)3-s + (0.843 − 0.537i)4-s + (−0.315 − 0.948i)5-s + (−0.985 − 0.168i)6-s + (−0.824 + 0.565i)7-s + (0.659 − 0.752i)8-s + (0.617 + 0.786i)9-s + (−0.568 − 0.822i)10-s + (0.124 − 0.992i)11-s + (−0.993 + 0.114i)12-s + (−0.767 + 0.641i)13-s + (−0.633 + 0.773i)14-s + (−0.131 + 0.991i)15-s + (0.422 − 0.906i)16-s + (0.708 − 0.705i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.849 + 0.527i$
Analytic conductor: \(4.46286\)
Root analytic conductor: \(4.46286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 961,\ (0:\ ),\ -0.849 + 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1990989081 - 0.6982634444i\)
\(L(\frac12)\) \(\approx\) \(-0.1990989081 - 0.6982634444i\)
\(L(1)\) \(\approx\) \(0.8334680346 - 0.5726277221i\)
\(L(1)\) \(\approx\) \(0.8334680346 - 0.5726277221i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (0.960 - 0.279i)T \)
3 \( 1 + (-0.899 - 0.437i)T \)
5 \( 1 + (-0.315 - 0.948i)T \)
7 \( 1 + (-0.824 + 0.565i)T \)
11 \( 1 + (0.124 - 0.992i)T \)
13 \( 1 + (-0.767 + 0.641i)T \)
17 \( 1 + (0.708 - 0.705i)T \)
19 \( 1 + (-0.867 + 0.497i)T \)
23 \( 1 + (0.111 - 0.993i)T \)
29 \( 1 + (-0.910 - 0.412i)T \)
37 \( 1 + (-0.557 - 0.830i)T \)
41 \( 1 + (-0.488 + 0.872i)T \)
43 \( 1 + (-0.958 - 0.286i)T \)
47 \( 1 + (-0.674 + 0.738i)T \)
53 \( 1 + (-0.340 + 0.940i)T \)
59 \( 1 + (0.938 + 0.344i)T \)
61 \( 1 + (-0.954 + 0.299i)T \)
67 \( 1 + (-0.839 + 0.543i)T \)
71 \( 1 + (0.994 - 0.107i)T \)
73 \( 1 + (0.835 + 0.548i)T \)
79 \( 1 + (0.835 - 0.548i)T \)
83 \( 1 + (-0.722 - 0.691i)T \)
89 \( 1 + (-0.476 + 0.879i)T \)
97 \( 1 + (-0.998 + 0.0607i)T \)
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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

−22.397445299207383405375287221373, −21.81001359353077807364331217561, −20.93331140441809943704287321523, −19.916364764432087747585595525523, −19.319425135594606175096853543729, −18.06586314493075897319288765067, −17.12484397210399236881630005924, −16.8115456918501699701137583767, −15.566751938785199918632203500067, −15.20647913640509104783048952620, −14.55447878378577669193836460061, −13.32977205045223920500329072877, −12.59565880928424450737452180243, −11.947915160071995728099415964, −11.00378326954880604893194657834, −10.31309592576134565656804836596, −9.70398959287284604192100658960, −7.91801679793365938936583014381, −6.96675493562192633385512702049, −6.69431842783662034132657871406, −5.61700954831170640314668064814, −4.7783612514405472483022812659, −3.74424545403704972405461532209, −3.27739069710920686582977815683, −1.87973709740140510804806243631, 0.2391880315869124372819899717, 1.49319545325665018920610586644, 2.57744937118011498121448089320, 3.77402142166107791597395209357, 4.71005168546099419404210538842, 5.49035531545962349150891091434, 6.15970235883801129680641120067, 6.97733798110599979136384508487, 8.04708731592609406652334437206, 9.25089248687758278919394891635, 10.15540879411799454254324101735, 11.21054681362306691022348436220, 11.93835213201518392620255531454, 12.42594613417600753391196297231, 13.05934022637711753907911044452, 13.87137099467692813557728524260, 14.92891136547221457982547477116, 15.942252910054476408353436718309, 16.663856746074900625114333163599, 16.758226458652100484929183035257, 18.48297886566647141314385851677, 19.10875018569987049933832719178, 19.594977592512591371853222020640, 20.7739513323920286729065086785, 21.46249431483284346630916246578

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