↑

Properties

Label 1-967-967.714-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

Related objects

Downloads

Learn more

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: $\chi_{967} (714, \cdot )$
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 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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