Properties

Label 1-967-967.795-r0-0-0
Degree $1$
Conductor $967$
Sign $0.578 - 0.815i$
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  + (−0.334 − 0.942i)2-s + (−0.990 + 0.136i)3-s + (−0.775 + 0.631i)4-s + (0.203 − 0.979i)5-s + (0.460 + 0.887i)6-s + (−0.917 + 0.398i)7-s + (0.854 + 0.519i)8-s + (0.962 − 0.269i)9-s + (−0.990 + 0.136i)10-s + (0.962 − 0.269i)11-s + (0.682 − 0.730i)12-s + (0.962 + 0.269i)13-s + (0.682 + 0.730i)14-s + (−0.0682 + 0.997i)15-s + (0.203 − 0.979i)16-s + (−0.0682 − 0.997i)17-s + ⋯
L(s)  = 1  + (−0.334 − 0.942i)2-s + (−0.990 + 0.136i)3-s + (−0.775 + 0.631i)4-s + (0.203 − 0.979i)5-s + (0.460 + 0.887i)6-s + (−0.917 + 0.398i)7-s + (0.854 + 0.519i)8-s + (0.962 − 0.269i)9-s + (−0.990 + 0.136i)10-s + (0.962 − 0.269i)11-s + (0.682 − 0.730i)12-s + (0.962 + 0.269i)13-s + (0.682 + 0.730i)14-s + (−0.0682 + 0.997i)15-s + (0.203 − 0.979i)16-s + (−0.0682 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (795, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.578 - 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6705375209 - 0.3464746864i\)
\(L(\frac12)\) \(\approx\) \(0.6705375209 - 0.3464746864i\)
\(L(1)\) \(\approx\) \(0.5899469307 - 0.2767015209i\)
\(L(1)\) \(\approx\) \(0.5899469307 - 0.2767015209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.334 - 0.942i)T \)
3 \( 1 + (-0.990 + 0.136i)T \)
5 \( 1 + (0.203 - 0.979i)T \)
7 \( 1 + (-0.917 + 0.398i)T \)
11 \( 1 + (0.962 - 0.269i)T \)
13 \( 1 + (0.962 + 0.269i)T \)
17 \( 1 + (-0.0682 - 0.997i)T \)
19 \( 1 + (-0.990 + 0.136i)T \)
23 \( 1 + (-0.775 + 0.631i)T \)
29 \( 1 + (0.962 + 0.269i)T \)
31 \( 1 + (0.854 + 0.519i)T \)
37 \( 1 + (0.962 + 0.269i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (-0.0682 + 0.997i)T \)
47 \( 1 + (-0.0682 + 0.997i)T \)
53 \( 1 + (-0.775 + 0.631i)T \)
59 \( 1 + (-0.775 - 0.631i)T \)
61 \( 1 + (0.460 + 0.887i)T \)
67 \( 1 + (0.854 - 0.519i)T \)
71 \( 1 + (-0.334 + 0.942i)T \)
73 \( 1 + (-0.775 - 0.631i)T \)
79 \( 1 + (-0.775 - 0.631i)T \)
83 \( 1 + (0.460 + 0.887i)T \)
89 \( 1 + (0.854 - 0.519i)T \)
97 \( 1 + T \)
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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

−22.15002631556503605708028652256, −21.56078542974506464172934387608, −19.983223070184540243680118484308, −19.058892051940986248639919755232, −18.71257241975616919634387425733, −17.56996476275969213409249679178, −17.332145814444002289587779520735, −16.40900590541715287271398667405, −15.67075021938866632168512733493, −14.94974904254645771565418889991, −13.97140682756054475489251169139, −13.22966210612388079422661465192, −12.38096531568650406807304934148, −11.16619420003689445894560984720, −10.33790419338504074033631899921, −9.996798376332211462143561492609, −8.78922439109645751921576436223, −7.7296186311264335578925802543, −6.654321722314083573079470957773, −6.39576723171137843188475732986, −5.84881621074669176256322740633, −4.31850497763842694543287320832, −3.762790829142983867889307706209, −1.97871536760824313082374625368, −0.65448937454861296226829005197, 0.81368347567990370450214533046, 1.57712117340379167898915373820, 3.00663359780204669014434995056, 4.13244107537274069261820103213, 4.69831950748916002438814852942, 5.94774786505728076824215991345, 6.50579491693700773734898103935, 8.00501163051954418375917511122, 9.01346916206249380526103835196, 9.49653067449385772097161174992, 10.28664290969417862917642727766, 11.40772854479838322365484275959, 11.85630221191604617360338416120, 12.665254931341491865718667267710, 13.24555787186785626441004330880, 14.15228547535955747505703934383, 15.85839760993923494725706741143, 16.21597443929431163479727791414, 17.002200286637038339412779832850, 17.719315618136757733837571859659, 18.48736943107101360512245624805, 19.344089525700684343253991177029, 19.994477407355875402586489229089, 21.008165422816837396008553648384, 21.574162460227891667844643131531

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