Properties

Label 1-967-967.154-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.822 - 0.569i$
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  + (−0.967 − 0.250i)2-s + (0.932 + 0.362i)3-s + (0.874 + 0.485i)4-s + (−0.710 + 0.703i)5-s + (−0.811 − 0.584i)6-s + (0.442 + 0.896i)7-s + (−0.724 − 0.689i)8-s + (0.737 + 0.675i)9-s + (0.864 − 0.502i)10-s + (−0.371 − 0.928i)11-s + (0.638 + 0.769i)12-s + (0.371 − 0.928i)13-s + (−0.203 − 0.979i)14-s + (−0.917 + 0.398i)15-s + (0.527 + 0.849i)16-s + (0.653 + 0.756i)17-s + ⋯
L(s)  = 1  + (−0.967 − 0.250i)2-s + (0.932 + 0.362i)3-s + (0.874 + 0.485i)4-s + (−0.710 + 0.703i)5-s + (−0.811 − 0.584i)6-s + (0.442 + 0.896i)7-s + (−0.724 − 0.689i)8-s + (0.737 + 0.675i)9-s + (0.864 − 0.502i)10-s + (−0.371 − 0.928i)11-s + (0.638 + 0.769i)12-s + (0.371 − 0.928i)13-s + (−0.203 − 0.979i)14-s + (−0.917 + 0.398i)15-s + (0.527 + 0.849i)16-s + (0.653 + 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.822 - 0.569i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.822 - 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04722582896 + 0.1512397592i\)
\(L(\frac12)\) \(\approx\) \(-0.04722582896 + 0.1512397592i\)
\(L(1)\) \(\approx\) \(0.7510238834 + 0.1767178794i\)
\(L(1)\) \(\approx\) \(0.7510238834 + 0.1767178794i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.967 - 0.250i)T \)
3 \( 1 + (0.932 + 0.362i)T \)
5 \( 1 + (-0.710 + 0.703i)T \)
7 \( 1 + (0.442 + 0.896i)T \)
11 \( 1 + (-0.371 - 0.928i)T \)
13 \( 1 + (0.371 - 0.928i)T \)
17 \( 1 + (0.653 + 0.756i)T \)
19 \( 1 + (-0.560 + 0.828i)T \)
23 \( 1 + (-0.165 - 0.986i)T \)
29 \( 1 + (-0.494 - 0.869i)T \)
31 \( 1 + (0.951 + 0.307i)T \)
37 \( 1 + (-0.987 - 0.155i)T \)
41 \( 1 + (-0.962 + 0.269i)T \)
43 \( 1 + (-0.999 - 0.0390i)T \)
47 \( 1 + (0.184 + 0.982i)T \)
53 \( 1 + (-0.576 - 0.816i)T \)
59 \( 1 + (0.924 + 0.380i)T \)
61 \( 1 + (0.962 - 0.269i)T \)
67 \( 1 + (-0.951 + 0.307i)T \)
71 \( 1 + (-0.967 + 0.250i)T \)
73 \( 1 + (-0.576 + 0.816i)T \)
79 \( 1 + (-0.279 - 0.960i)T \)
83 \( 1 + (-0.984 - 0.174i)T \)
89 \( 1 + (-0.951 + 0.307i)T \)
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

−20.60813142152585356353624436622, −20.28415038820018751317504019627, −19.47923286445265727551880375077, −18.872132031692817305781428358504, −17.979927764304483008705117258887, −17.19544984252982342583637484652, −16.39672697875585608890632551555, −15.547989955818694553629661289661, −14.968533828545830221132793642477, −13.97613838332286088467573965677, −13.221567976958811747471450598237, −12.06897456044640399423678629720, −11.47241201623929625350797765098, −10.29797617444763661203929241102, −9.52215591561437724948436180854, −8.7142638517285992627926812638, −8.06491674969105609300158311056, −7.153246391958391970618919287695, −6.96083606137229996813690061271, −5.18387326082720659500748259335, −4.25952648255129040717130423868, −3.19958748854810900382339145697, −1.85449833683585087509938174677, −1.23672057818778153476686801909, −0.03991582765998239597802160429, 1.49676880290505252863963816975, 2.611742978006654899263619230389, 3.18644215152652132464815121537, 4.040820087727471704591713455311, 5.612939198817456750383243649714, 6.58327802534760430635783069509, 7.8949641804776344068558003373, 8.232532802547336940089734781714, 8.68230381938603451981678315648, 10.12349365937289074032044737082, 10.436880068522528056456353966872, 11.37673658255352838494694687133, 12.23564704538012358848744024377, 13.13945332622118507992981480765, 14.47511230770716040099881970139, 14.99094793432979542215610967949, 15.729155678352064121091742954116, 16.29446155818294580418313477978, 17.4730530650254093407661045924, 18.507339430117716861217839490481, 18.937284350163716925353915155666, 19.3401896925259178588164878627, 20.514572477168159019075187366499, 20.96517336368114654806779614069, 21.73771767071403666620426640927

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