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Properties

Label 1-967-967.125-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.968 - 0.247i$
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  + (0.527 + 0.849i)2-s + (−0.0876 − 0.996i)3-s + (−0.442 + 0.896i)4-s + (0.999 + 0.0195i)5-s + (0.799 − 0.600i)6-s + (0.260 + 0.965i)7-s + (−0.995 + 0.0974i)8-s + (−0.984 + 0.174i)9-s + (0.511 + 0.859i)10-s + (0.0487 − 0.998i)11-s + (0.932 + 0.362i)12-s + (−0.0487 − 0.998i)13-s + (−0.682 + 0.730i)14-s + (−0.0682 − 0.997i)15-s + (−0.608 − 0.793i)16-s + (−0.957 − 0.288i)17-s + ⋯
L(s)  = 1  + (0.527 + 0.849i)2-s + (−0.0876 − 0.996i)3-s + (−0.442 + 0.896i)4-s + (0.999 + 0.0195i)5-s + (0.799 − 0.600i)6-s + (0.260 + 0.965i)7-s + (−0.995 + 0.0974i)8-s + (−0.984 + 0.174i)9-s + (0.511 + 0.859i)10-s + (0.0487 − 0.998i)11-s + (0.932 + 0.362i)12-s + (−0.0487 − 0.998i)13-s + (−0.682 + 0.730i)14-s + (−0.0682 − 0.997i)15-s + (−0.608 − 0.793i)16-s + (−0.957 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.968 - 0.247i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.968 - 0.247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05146420264 + 0.4088581313i\)
\(L(\frac12)\) \(\approx\) \(-0.05146420264 + 0.4088581313i\)
\(L(1)\) \(\approx\) \(1.147295793 + 0.3032591122i\)
\(L(1)\) \(\approx\) \(1.147295793 + 0.3032591122i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.527 + 0.849i)T \)
3 \( 1 + (-0.0876 - 0.996i)T \)
5 \( 1 + (0.999 + 0.0195i)T \)
7 \( 1 + (0.260 + 0.965i)T \)
11 \( 1 + (0.0487 - 0.998i)T \)
13 \( 1 + (-0.0487 - 0.998i)T \)
17 \( 1 + (-0.957 - 0.288i)T \)
19 \( 1 + (0.724 - 0.689i)T \)
23 \( 1 + (-0.787 + 0.615i)T \)
29 \( 1 + (0.477 + 0.878i)T \)
31 \( 1 + (0.316 + 0.948i)T \)
37 \( 1 + (-0.811 - 0.584i)T \)
41 \( 1 + (-0.460 + 0.887i)T \)
43 \( 1 + (-0.987 - 0.155i)T \)
47 \( 1 + (-0.737 + 0.675i)T \)
53 \( 1 + (-0.775 - 0.631i)T \)
59 \( 1 + (0.00975 + 0.999i)T \)
61 \( 1 + (0.460 - 0.887i)T \)
67 \( 1 + (-0.316 + 0.948i)T \)
71 \( 1 + (0.527 - 0.849i)T \)
73 \( 1 + (-0.775 + 0.631i)T \)
79 \( 1 + (-0.425 + 0.905i)T \)
83 \( 1 + (0.763 + 0.646i)T \)
89 \( 1 + (-0.316 + 0.948i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−21.03230425207304547043879173080, −20.48765639341430748289122721374, −20.08481616779436343749858640896, −18.9203209690481180531998676956, −17.839184501652927693004835480319, −17.31015798601803381528934396861, −16.44952373953633625159995953000, −15.3652774731946048483809386926, −14.53811607354313784276500976019, −13.9220604567919715542709096424, −13.357188890911013662343882935071, −12.1346648239073408809128989021, −11.43723198642615604046879546224, −10.42694931939378508595491212732, −10.00879854026705910025782335410, −9.416030472679127690732978079179, −8.38524398950427132374289254024, −6.77845070307605624998396708102, −6.02622479885280834735120741257, −4.85923021972042423091286540998, −4.43462371338385912407211203110, −3.57695801052767753583322571372, −2.29367398198157382125265230957, −1.58523859884070028726073766076, −0.0655393821931194227082704799, 1.37648535636730939052550891258, 2.64267796516160304808978445789, 3.17841777863770347301853244584, 5.13012741944658498248425370979, 5.40953923070618045577578606130, 6.32526294749194158302180041648, 6.908954658137416236066836333970, 8.14089378861980891561843234556, 8.61878488752018865087448833987, 9.48800160147823111255105076867, 10.969542863356979098426510277033, 11.82683678442833256959053401322, 12.64973541324252761648769917999, 13.38142442323287791950954701053, 13.9002534478576633811227201368, 14.65184965519805388415045453221, 15.67153360657428420551777515584, 16.376993424753704489905062668513, 17.6346841637993869269855464822, 17.79117053424643819619192910862, 18.397473535914728606633121597530, 19.53692886618883245734369822130, 20.51192484053418973349892755654, 21.63884205686414418608470708936, 22.00184154245961767873405581603

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