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Properties

Label 1-31e2-961.125-r0-0-0
Degree $1$
Conductor $961$
Sign $0.0734 - 0.997i$
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  + (−0.440 − 0.897i)2-s + (−0.874 + 0.485i)3-s + (−0.612 + 0.790i)4-s + (−0.994 − 0.101i)5-s + (0.820 + 0.571i)6-s + (0.918 − 0.394i)7-s + (0.979 + 0.201i)8-s + (0.528 − 0.848i)9-s + (0.347 + 0.937i)10-s + (−0.954 + 0.299i)11-s + (0.151 − 0.988i)12-s + (0.528 + 0.848i)13-s + (−0.758 − 0.651i)14-s + (0.918 − 0.394i)15-s + (−0.250 − 0.968i)16-s + (0.688 − 0.724i)17-s + ⋯
L(s)  = 1  + (−0.440 − 0.897i)2-s + (−0.874 + 0.485i)3-s + (−0.612 + 0.790i)4-s + (−0.994 − 0.101i)5-s + (0.820 + 0.571i)6-s + (0.918 − 0.394i)7-s + (0.979 + 0.201i)8-s + (0.528 − 0.848i)9-s + (0.347 + 0.937i)10-s + (−0.954 + 0.299i)11-s + (0.151 − 0.988i)12-s + (0.528 + 0.848i)13-s + (−0.758 − 0.651i)14-s + (0.918 − 0.394i)15-s + (−0.250 − 0.968i)16-s + (0.688 − 0.724i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0734 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0734 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.0734 - 0.997i$
Analytic conductor: \(4.46286\)
Root analytic conductor: \(4.46286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 961,\ (0:\ ),\ 0.0734 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4111129454 - 0.3819337293i\)
\(L(\frac12)\) \(\approx\) \(0.4111129454 - 0.3819337293i\)
\(L(1)\) \(\approx\) \(0.5231053448 - 0.1796824454i\)
\(L(1)\) \(\approx\) \(0.5231053448 - 0.1796824454i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.440 - 0.897i)T \)
3 \( 1 + (-0.874 + 0.485i)T \)
5 \( 1 + (-0.994 - 0.101i)T \)
7 \( 1 + (0.918 - 0.394i)T \)
11 \( 1 + (-0.954 + 0.299i)T \)
13 \( 1 + (0.528 + 0.848i)T \)
17 \( 1 + (0.688 - 0.724i)T \)
19 \( 1 + (-0.0506 - 0.998i)T \)
23 \( 1 + (-0.994 + 0.101i)T \)
29 \( 1 + (-0.994 + 0.101i)T \)
37 \( 1 + (0.528 + 0.848i)T \)
41 \( 1 + (0.979 + 0.201i)T \)
43 \( 1 + (0.347 - 0.937i)T \)
47 \( 1 + (-0.994 + 0.101i)T \)
53 \( 1 + (-0.874 + 0.485i)T \)
59 \( 1 + (0.528 + 0.848i)T \)
61 \( 1 + (0.151 + 0.988i)T \)
67 \( 1 + (0.688 - 0.724i)T \)
71 \( 1 + (-0.0506 + 0.998i)T \)
73 \( 1 + (-0.758 - 0.651i)T \)
79 \( 1 + (-0.758 + 0.651i)T \)
83 \( 1 + (-0.440 - 0.897i)T \)
89 \( 1 + (0.918 + 0.394i)T \)
97 \( 1 + (-0.612 - 0.790i)T \)
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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

−22.30953640531703422426737131811, −21.2534918655804662564067334129, −20.241226046842317520267505087633, −19.11034832443661694521825717020, −18.62775629396696364660750861088, −18.00453649459188028186213755256, −17.34065283728218737005347767070, −16.16589731052388489783236316299, −16.01302350842594569292605172187, −14.92147769849181956337187869930, −14.31089747494810250377063802668, −13.02411899038548513472362766699, −12.451865301233834249922914364980, −11.25635033795922374398234078208, −10.813821940733701685294599763451, −9.91293957766007194389256627067, −8.30994093477847182861497938457, −7.98290579442315566241615700714, −7.48599903351154453493874888616, −6.09549047753139876345160083019, −5.648144337086012382251817630327, −4.76435899252880694308865755103, −3.71841988618507016379642086154, −1.95692459178680000652542483084, −0.82185266085818603081202685050, 0.4831062658018851101358397891, 1.59622049604944516471624192393, 2.9928042723881655853953975201, 4.14204414624103219059091480458, 4.53841787510425620789795862858, 5.46278388823931870443582428919, 7.09016394453536428889245043271, 7.70103279029729277578666569941, 8.64659241150583548614078409080, 9.61248872668507464120219391498, 10.48263657799439283169565340989, 11.32729383983212275724477513049, 11.51590715784763359477860849280, 12.42448866959073033877345957271, 13.34277741265617121607923698834, 14.40520483471054919153833690187, 15.49126440193584594172401150717, 16.2517199886350578431517963921, 16.86162332214826459623512081133, 17.8811832654831525838141816408, 18.33035690374731335417838304324, 19.14629536113328227458076345777, 20.31548735896572054407121563951, 20.68110169137963566944893804131, 21.41532259969089350315051758127

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