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Properties

Label 1-967-967.452-r0-0-0
Degree $1$
Conductor $967$
Sign $0.907 + 0.419i$
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  + (−0.555 − 0.831i)2-s + (−0.984 + 0.174i)3-s + (−0.383 + 0.923i)4-s + (−0.465 − 0.884i)5-s + (0.692 + 0.721i)6-s + (0.867 + 0.497i)7-s + (0.981 − 0.193i)8-s + (0.938 − 0.344i)9-s + (−0.477 + 0.878i)10-s + (−0.995 − 0.0974i)11-s + (0.216 − 0.976i)12-s + (0.581 + 0.813i)13-s + (−0.0682 − 0.997i)14-s + (0.613 + 0.789i)15-s + (−0.705 − 0.708i)16-s + (0.833 + 0.552i)17-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)2-s + (−0.984 + 0.174i)3-s + (−0.383 + 0.923i)4-s + (−0.465 − 0.884i)5-s + (0.692 + 0.721i)6-s + (0.867 + 0.497i)7-s + (0.981 − 0.193i)8-s + (0.938 − 0.344i)9-s + (−0.477 + 0.878i)10-s + (−0.995 − 0.0974i)11-s + (0.216 − 0.976i)12-s + (0.581 + 0.813i)13-s + (−0.0682 − 0.997i)14-s + (0.613 + 0.789i)15-s + (−0.705 − 0.708i)16-s + (0.833 + 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.907 + 0.419i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (452, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.907 + 0.419i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6042620525 + 0.1327324873i\)
\(L(\frac12)\) \(\approx\) \(0.6042620525 + 0.1327324873i\)
\(L(1)\) \(\approx\) \(0.5771627528 - 0.1184509988i\)
\(L(1)\) \(\approx\) \(0.5771627528 - 0.1184509988i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.555 - 0.831i)T \)
3 \( 1 + (-0.984 + 0.174i)T \)
5 \( 1 + (-0.465 - 0.884i)T \)
7 \( 1 + (0.867 + 0.497i)T \)
11 \( 1 + (-0.995 - 0.0974i)T \)
13 \( 1 + (0.581 + 0.813i)T \)
17 \( 1 + (0.833 + 0.552i)T \)
19 \( 1 + (0.840 + 0.541i)T \)
23 \( 1 + (0.241 - 0.970i)T \)
29 \( 1 + (-0.544 + 0.838i)T \)
31 \( 1 + (0.919 + 0.392i)T \)
37 \( 1 + (0.663 - 0.748i)T \)
41 \( 1 + (-0.576 - 0.816i)T \)
43 \( 1 + (-0.741 + 0.670i)T \)
47 \( 1 + (-0.906 + 0.422i)T \)
53 \( 1 + (-0.949 - 0.313i)T \)
59 \( 1 + (0.516 + 0.856i)T \)
61 \( 1 + (-0.419 + 0.907i)T \)
67 \( 1 + (-0.799 - 0.600i)T \)
71 \( 1 + (-0.442 - 0.896i)T \)
73 \( 1 + (-0.949 + 0.313i)T \)
79 \( 1 + (-0.347 + 0.937i)T \)
83 \( 1 + (-0.936 - 0.350i)T \)
89 \( 1 + (-0.120 + 0.992i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−21.994617587595588754570459475573, −20.87909399395894032809858867750, −20.00072856482364119773814626443, −18.757825381053914129377045731212, −18.47194017034182053044250244818, −17.71072521809520933588059127057, −17.18023449605283355289676353817, −16.09854393655779147987048814571, −15.52860378491201679364470924559, −14.88290573259407178415969098385, −13.729884159949715073521409096513, −13.205793973950318828863259604715, −11.52041514533056227684990651191, −11.34049912605162183444408084160, −10.234722919034797766364070279344, −9.90561359908083423757389743797, −8.14207566683209410987279041176, −7.6846440482326351394256709924, −7.09239047846385616812085082256, −6.038975673133534593259509447856, −5.27068481708522507371420332437, −4.54063173007182061038674505709, −3.14373746334543367971721379102, −1.56143544833208693500127692362, −0.46942107973614743883650576326, 1.06858945675655106726107156269, 1.73933574440430184663844037651, 3.26220681909661743809320449387, 4.354207981152319971461551415875, 4.96979431304719911531834967648, 5.84135553507428765486033808190, 7.31968453505550143663259727920, 8.12684877358558678342389082913, 8.80842478961719564049578581887, 9.80236238735252354064102233924, 10.68911556946965467446879027182, 11.36793290788388132371959085904, 12.07093190977533712719843676806, 12.57956063813535785016686329499, 13.47168151327543526045200058560, 14.73286189088682406548308938665, 15.98903926524754112727570638310, 16.35782904430134172694281827921, 17.1144063761098227619718478010, 18.09773820249925345943043030651, 18.48864813887636643672604033850, 19.30583341275738318771039292709, 20.54364887578571423482099210008, 21.0452346863916903548061314874, 21.40239310866615617177131522634

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