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Properties

Label 1-967-967.12-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.464 + 0.885i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (−0.618 + 0.785i)2-s + (−0.527 − 0.849i)3-s + (−0.235 − 0.971i)4-s + (0.992 + 0.123i)5-s + (0.993 + 0.110i)6-s + (−0.413 + 0.910i)7-s + (0.909 + 0.416i)8-s + (−0.442 + 0.896i)9-s + (−0.710 + 0.703i)10-s + (−0.977 + 0.212i)11-s + (−0.701 + 0.712i)12-s + (−0.672 + 0.739i)13-s + (−0.460 − 0.887i)14-s + (−0.419 − 0.907i)15-s + (−0.889 + 0.457i)16-s + (0.279 − 0.960i)17-s + ⋯
L(s)  = 1  + (−0.618 + 0.785i)2-s + (−0.527 − 0.849i)3-s + (−0.235 − 0.971i)4-s + (0.992 + 0.123i)5-s + (0.993 + 0.110i)6-s + (−0.413 + 0.910i)7-s + (0.909 + 0.416i)8-s + (−0.442 + 0.896i)9-s + (−0.710 + 0.703i)10-s + (−0.977 + 0.212i)11-s + (−0.701 + 0.712i)12-s + (−0.672 + 0.739i)13-s + (−0.460 − 0.887i)14-s + (−0.419 − 0.907i)15-s + (−0.889 + 0.457i)16-s + (0.279 − 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.464 + 0.885i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.464 + 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4487764920 + 0.7417657670i\)
\(L(\frac12)\) \(\approx\) \(0.4487764920 + 0.7417657670i\)
\(L(1)\) \(\approx\) \(0.6537788041 + 0.1818254711i\)
\(L(1)\) \(\approx\) \(0.6537788041 + 0.1818254711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.618 + 0.785i)T \)
3 \( 1 + (-0.527 - 0.849i)T \)
5 \( 1 + (0.992 + 0.123i)T \)
7 \( 1 + (-0.413 + 0.910i)T \)
11 \( 1 + (-0.977 + 0.212i)T \)
13 \( 1 + (-0.672 + 0.739i)T \)
17 \( 1 + (0.279 - 0.960i)T \)
19 \( 1 + (0.807 - 0.589i)T \)
23 \( 1 + (0.511 + 0.859i)T \)
29 \( 1 + (-0.00975 - 0.999i)T \)
31 \( 1 + (-0.0552 + 0.998i)T \)
37 \( 1 + (0.285 - 0.958i)T \)
41 \( 1 + (0.917 + 0.398i)T \)
43 \( 1 + (-0.549 - 0.835i)T \)
47 \( 1 + (0.0162 - 0.999i)T \)
53 \( 1 + (0.377 + 0.926i)T \)
59 \( 1 + (0.0617 + 0.998i)T \)
61 \( 1 + (0.803 - 0.595i)T \)
67 \( 1 + (-0.892 - 0.451i)T \)
71 \( 1 + (-0.371 + 0.928i)T \)
73 \( 1 + (0.377 - 0.926i)T \)
79 \( 1 + (0.986 + 0.161i)T \)
83 \( 1 + (-0.705 + 0.708i)T \)
89 \( 1 + (0.837 - 0.547i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−20.99008830919748541665657307751, −20.72432283347387315775421404509, −19.975298375760250806429055844643, −18.902708854314541643990419297330, −17.95012031252354938288356189607, −17.456914448098220962227179490736, −16.56856908768657400635787799087, −16.33525357527314673102608855502, −14.97461579379177554496108161738, −14.02547837672229695194545339968, −12.90341681930308028252994110093, −12.65367993186278703567795201186, −11.29511803361070243857782354669, −10.498466137333004687587714338774, −10.11494249525954734019345759850, −9.55746229201367396690731663531, −8.4605195209110473909599338935, −7.523099298968697409418544853058, −6.34746014439288331638798210485, −5.347345042906599410105432546054, −4.51336643048603561634568582218, −3.385587728911880194285872428745, −2.69415561675690692125635010819, −1.24240147452498430182173738945, −0.318121255992524007451549341335, 0.87439113362201263081063628608, 2.09065370272397912218078592543, 2.67345076531540934333759607943, 5.07732321969300526675041010470, 5.31817329196416852233145240663, 6.21397536278308466626989097811, 7.07861581561250229136450232202, 7.60039558001371100605346128814, 8.88912537345694311050297430149, 9.486974614941384040736242259833, 10.29583980505135825675008445946, 11.34405137354089450586223802397, 12.20912353914552845047878670864, 13.3051588572514849766040989213, 13.741601978087172546105538714189, 14.701109062978090905965553332, 15.71620033729093757884190138412, 16.39429844616733388721750034682, 17.23879042313389214032767848425, 18.003239227280752948694701643174, 18.34906356258424331990402270898, 19.103781605428932925995140439299, 19.872967062090822370289147590370, 21.181516213025986483066628944832, 21.96374234805060242830272119371

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