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Properties

Label 1-967-967.144-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.981 - 0.190i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (−0.235 − 0.971i)2-s + (−0.442 + 0.896i)3-s + (−0.889 + 0.457i)4-s + (0.969 + 0.244i)5-s + (0.975 + 0.219i)6-s + (−0.658 − 0.752i)7-s + (0.653 + 0.756i)8-s + (−0.608 − 0.793i)9-s + (0.00975 − 0.999i)10-s + (0.909 − 0.416i)11-s + (−0.0162 − 0.999i)12-s + (−0.0941 − 0.995i)13-s + (−0.576 + 0.816i)14-s + (−0.648 + 0.761i)15-s + (0.581 − 0.813i)16-s + (−0.844 − 0.536i)17-s + ⋯
L(s)  = 1  + (−0.235 − 0.971i)2-s + (−0.442 + 0.896i)3-s + (−0.889 + 0.457i)4-s + (0.969 + 0.244i)5-s + (0.975 + 0.219i)6-s + (−0.658 − 0.752i)7-s + (0.653 + 0.756i)8-s + (−0.608 − 0.793i)9-s + (0.00975 − 0.999i)10-s + (0.909 − 0.416i)11-s + (−0.0162 − 0.999i)12-s + (−0.0941 − 0.995i)13-s + (−0.576 + 0.816i)14-s + (−0.648 + 0.761i)15-s + (0.581 − 0.813i)16-s + (−0.844 − 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.981 - 0.190i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.981 - 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04727751902 - 0.4929804037i\)
\(L(\frac12)\) \(\approx\) \(0.04727751902 - 0.4929804037i\)
\(L(1)\) \(\approx\) \(0.6401611046 - 0.2690484513i\)
\(L(1)\) \(\approx\) \(0.6401611046 - 0.2690484513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.235 - 0.971i)T \)
3 \( 1 + (-0.442 + 0.896i)T \)
5 \( 1 + (0.969 + 0.244i)T \)
7 \( 1 + (-0.658 - 0.752i)T \)
11 \( 1 + (0.909 - 0.416i)T \)
13 \( 1 + (-0.0941 - 0.995i)T \)
17 \( 1 + (-0.844 - 0.536i)T \)
19 \( 1 + (0.304 - 0.952i)T \)
23 \( 1 + (-0.477 + 0.878i)T \)
29 \( 1 + (-0.999 + 0.0195i)T \)
31 \( 1 + (-0.993 - 0.110i)T \)
37 \( 1 + (-0.837 - 0.547i)T \)
41 \( 1 + (0.682 + 0.730i)T \)
43 \( 1 + (-0.395 + 0.918i)T \)
47 \( 1 + (-0.999 - 0.0325i)T \)
53 \( 1 + (-0.715 + 0.699i)T \)
59 \( 1 + (-0.992 + 0.123i)T \)
61 \( 1 + (0.291 - 0.956i)T \)
67 \( 1 + (0.592 + 0.805i)T \)
71 \( 1 + (-0.724 - 0.689i)T \)
73 \( 1 + (-0.715 - 0.699i)T \)
79 \( 1 + (0.947 + 0.319i)T \)
83 \( 1 + (-0.00325 - 0.999i)T \)
89 \( 1 + (0.401 - 0.915i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−22.26556679870371795981809320746, −21.82987713768212955353046917948, −20.40072583483295339773435161704, −19.36587999253724639091472765537, −18.743967875257183608616024841582, −18.09875295805376914546442339228, −17.34227301990105475222636782663, −16.68179381494611326272277463801, −16.15490868001193347458191688537, −14.8321991241569011082112620276, −14.221890923573058068179376181737, −13.42053272756150254523880645901, −12.64930817951613151262848845041, −12.03097723912664995068880759669, −10.71178334665897561481097881241, −9.64939981926904585378014399758, −9.04038336380240645872479142011, −8.299781799758629521998716240390, −6.980582159190016103454086969670, −6.52150111141345529160372545086, −5.87377656571737314601600472223, −5.09176365115333202778467108533, −3.889728725269797253143708911166, −2.10053463022522298008141601016, −1.5346108364455327318117127114, 0.24469546390837709817898838561, 1.51380966472372724402358643671, 2.94004322752942522121019631172, 3.48292891456634147127021706734, 4.5114235760255380375601581264, 5.43971985890275603263599250412, 6.33923677516714909811163350458, 7.44969742482811582445919567983, 8.9490535153249159091905564532, 9.46557198683146828671260760825, 9.99614604233446868684541284482, 11.03915010288325272549124464768, 11.22334187108347252163797505684, 12.568737461930091304180012054843, 13.348272643181164345085206335647, 14.00194311214741439369615258119, 14.930998988462690480980049945665, 16.12189880700983627087635913468, 16.82513189474434983450504116567, 17.683053172756547239484097154993, 17.87980553786160762806763842471, 19.24410760596415816285339065053, 20.12686244298441407110384704548, 20.39999344805884413228523573434, 21.586699044758781508451571875488

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