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Properties

Label 1-967-967.35-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.328 + 0.944i$
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  + (−0.465 + 0.884i)2-s + (−0.407 − 0.913i)3-s + (−0.566 − 0.824i)4-s + (0.903 + 0.428i)5-s + (0.997 + 0.0649i)6-s + (−0.209 − 0.977i)7-s + (0.993 − 0.116i)8-s + (−0.668 + 0.744i)9-s + (−0.799 + 0.600i)10-s + (−0.998 − 0.0585i)11-s + (−0.522 + 0.852i)12-s + (0.549 + 0.835i)13-s + (0.962 + 0.269i)14-s + (0.0227 − 0.999i)15-s + (−0.359 + 0.933i)16-s + (0.938 + 0.344i)17-s + ⋯
L(s)  = 1  + (−0.465 + 0.884i)2-s + (−0.407 − 0.913i)3-s + (−0.566 − 0.824i)4-s + (0.903 + 0.428i)5-s + (0.997 + 0.0649i)6-s + (−0.209 − 0.977i)7-s + (0.993 − 0.116i)8-s + (−0.668 + 0.744i)9-s + (−0.799 + 0.600i)10-s + (−0.998 − 0.0585i)11-s + (−0.522 + 0.852i)12-s + (0.549 + 0.835i)13-s + (0.962 + 0.269i)14-s + (0.0227 − 0.999i)15-s + (−0.359 + 0.933i)16-s + (0.938 + 0.344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.328 + 0.944i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3644344015 + 0.5127009186i\)
\(L(\frac12)\) \(\approx\) \(0.3644344015 + 0.5127009186i\)
\(L(1)\) \(\approx\) \(0.6650628693 + 0.1612726552i\)
\(L(1)\) \(\approx\) \(0.6650628693 + 0.1612726552i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.465 + 0.884i)T \)
3 \( 1 + (-0.407 - 0.913i)T \)
5 \( 1 + (0.903 + 0.428i)T \)
7 \( 1 + (-0.209 - 0.977i)T \)
11 \( 1 + (-0.998 - 0.0585i)T \)
13 \( 1 + (0.549 + 0.835i)T \)
17 \( 1 + (0.938 + 0.344i)T \)
19 \( 1 + (-0.851 - 0.525i)T \)
23 \( 1 + (-0.145 + 0.989i)T \)
29 \( 1 + (0.279 + 0.960i)T \)
31 \( 1 + (-0.999 - 0.0325i)T \)
37 \( 1 + (-0.857 - 0.514i)T \)
41 \( 1 + (-0.775 + 0.631i)T \)
43 \( 1 + (-0.922 - 0.386i)T \)
47 \( 1 + (-0.454 + 0.890i)T \)
53 \( 1 + (-0.974 + 0.225i)T \)
59 \( 1 + (0.975 - 0.219i)T \)
61 \( 1 + (0.934 + 0.356i)T \)
67 \( 1 + (0.527 + 0.849i)T \)
71 \( 1 + (0.999 + 0.0390i)T \)
73 \( 1 + (-0.974 - 0.225i)T \)
79 \( 1 + (-0.00325 + 0.999i)T \)
83 \( 1 + (0.672 - 0.739i)T \)
89 \( 1 + (0.471 - 0.881i)T \)
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

−21.34488159228806217806145224167, −20.749929277043667999410980566569, −20.41062595949002826465556219658, −19.00454891708405582425466721111, −18.31240147600607262762174635527, −17.7375418305614164250502750423, −16.80966897458911004303998198711, −16.23414515177128977716286476182, −15.32775257734223732095761828440, −14.33022521203062036428268654705, −13.204152746628762006509570840530, −12.56722253522508222382067476183, −11.867876149831083970251660672829, −10.758905528591909553476255747083, −10.147580066712175495428680406503, −9.66243632751877988969937997696, −8.57890819366637548430183573107, −8.22717873563317681393530416526, −6.412875252667733587338707169145, −5.40561902099725739497466189496, −4.99500727972588478666786297418, −3.64619443165221733074150054572, −2.796724034866715165707414603, −1.88840541831334154600401861043, −0.35169084214505081664322893608, 1.215455435698108027704639289273, 1.96925570553732619634177354255, 3.46987851370071426047439103052, 4.96644077695991422250173489021, 5.66934697166362754627712946349, 6.573656540400280019130918166459, 7.04965325057256448339586418305, 7.88000116768719976070607976769, 8.80656286627739771793556924226, 9.9045910233011153201993630083, 10.5996275917449980247993575373, 11.24070604514085623606697184121, 12.85935059049913070964870940667, 13.312046437754046299939039705491, 14.05254598100988652149830932218, 14.64126869515933971634558487827, 15.98568749349589868704506867044, 16.633562974613376124458528158028, 17.36819752050484082719927045474, 17.88486312546061479922639113179, 18.73948352009817676112811359356, 19.17071933146648377795999196001, 20.209050006923294528357312422853, 21.343258484393292016492973165048, 22.16593489395522809199172594841

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