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Properties

Label 1-967-967.123-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.923 + 0.383i$
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  + (−0.844 + 0.536i)2-s + (0.833 − 0.552i)3-s + (0.425 − 0.905i)4-s + (−0.608 + 0.793i)5-s + (−0.407 + 0.913i)6-s + (−0.184 − 0.982i)7-s + (0.126 + 0.991i)8-s + (0.389 − 0.921i)9-s + (0.0876 − 0.996i)10-s + (−0.750 + 0.660i)11-s + (−0.145 − 0.989i)12-s + (−0.750 − 0.660i)13-s + (0.682 + 0.730i)14-s + (−0.0682 + 0.997i)15-s + (−0.638 − 0.769i)16-s + (−0.371 + 0.928i)17-s + ⋯
L(s)  = 1  + (−0.844 + 0.536i)2-s + (0.833 − 0.552i)3-s + (0.425 − 0.905i)4-s + (−0.608 + 0.793i)5-s + (−0.407 + 0.913i)6-s + (−0.184 − 0.982i)7-s + (0.126 + 0.991i)8-s + (0.389 − 0.921i)9-s + (0.0876 − 0.996i)10-s + (−0.750 + 0.660i)11-s + (−0.145 − 0.989i)12-s + (−0.750 − 0.660i)13-s + (0.682 + 0.730i)14-s + (−0.0682 + 0.997i)15-s + (−0.638 − 0.769i)16-s + (−0.371 + 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.923 + 0.383i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.923 + 0.383i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04014402110 + 0.2015878496i\)
\(L(\frac12)\) \(\approx\) \(0.04014402110 + 0.2015878496i\)
\(L(1)\) \(\approx\) \(0.6285798070 + 0.06524644267i\)
\(L(1)\) \(\approx\) \(0.6285798070 + 0.06524644267i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.844 + 0.536i)T \)
3 \( 1 + (0.833 - 0.552i)T \)
5 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 + (-0.184 - 0.982i)T \)
11 \( 1 + (-0.750 + 0.660i)T \)
13 \( 1 + (-0.750 - 0.660i)T \)
17 \( 1 + (-0.371 + 0.928i)T \)
19 \( 1 + (0.353 + 0.935i)T \)
23 \( 1 + (0.972 - 0.232i)T \)
29 \( 1 + (-0.984 + 0.174i)T \)
31 \( 1 + (-0.544 - 0.838i)T \)
37 \( 1 + (-0.477 + 0.878i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (0.494 - 0.869i)T \)
47 \( 1 + (-0.957 - 0.288i)T \)
53 \( 1 + (-0.775 + 0.631i)T \)
59 \( 1 + (-0.442 + 0.896i)T \)
61 \( 1 + (0.460 + 0.887i)T \)
67 \( 1 + (-0.544 + 0.838i)T \)
71 \( 1 + (-0.844 - 0.536i)T \)
73 \( 1 + (-0.775 - 0.631i)T \)
79 \( 1 + (-0.977 + 0.212i)T \)
83 \( 1 + (-0.0292 - 0.999i)T \)
89 \( 1 + (-0.544 + 0.838i)T \)
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

−21.19026026351008971697990911932, −20.60574196372464839869356043694, −19.69463627999127132078771824245, −19.22586093076194055810093958089, −18.59329925814056121178118589533, −17.53568083621741038613490353222, −16.4704724944588957791827359756, −15.89629844288788379801518132096, −15.49999836618126563833437466598, −14.305931930991635471224411418209, −13.10492651443415400724850277220, −12.65733370695048388767419659552, −11.46332516523366670959990616324, −11.06809715290185424809542016665, −9.68661270991266880284936716634, −9.120158256100755991751521416728, −8.73800236699536556044643044668, −7.74261442665885745346494229503, −7.07644140763953014119561145238, −5.31426280575025830837400999730, −4.599016154806674415887182002840, −3.35882468543718778849490335084, −2.73317338141329532426157313632, −1.76785715305490816221752106154, −0.10196095290891936589980707858, 1.375004190805797889446635986759, 2.46300761288610088003891977476, 3.38455077627208361188823389050, 4.51662271880588305819213629227, 5.94538135434325175887651162406, 6.9194858843440165229999732511, 7.60479375465628196927622967037, 7.799310591784418224564716123140, 8.95171276970415893949134237653, 10.11656471037674070009423026596, 10.3671588663017886676765790455, 11.48087041530306539503880193260, 12.663452871230832160413688086, 13.416843064514900644346495946326, 14.614227169310695821568290607525, 14.85286510277713970651059837844, 15.60442419439310957202190100438, 16.68318662383451955927726495465, 17.51878788953827194189771107408, 18.24926350559177961362123123984, 18.96909240332969857516353421910, 19.54905753312268488552634202044, 20.30195521332910610186908882167, 20.76016614646247225245589460505, 22.39521532847337445447834283324

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