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Properties

Label 1-967-967.160-r1-0-0
Degree $1$
Conductor $967$
Sign $0.972 - 0.234i$
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  + (−0.741 + 0.670i)2-s + (0.977 + 0.212i)3-s + (0.100 − 0.994i)4-s + (0.922 − 0.386i)5-s + (−0.867 + 0.497i)6-s + (0.120 − 0.992i)7-s + (0.592 + 0.805i)8-s + (0.909 + 0.416i)9-s + (−0.425 + 0.905i)10-s + (0.892 − 0.451i)11-s + (0.310 − 0.950i)12-s + (0.0552 + 0.998i)13-s + (0.576 + 0.816i)14-s + (0.983 − 0.181i)15-s + (−0.979 − 0.200i)16-s + (−0.945 − 0.325i)17-s + ⋯
L(s)  = 1  + (−0.741 + 0.670i)2-s + (0.977 + 0.212i)3-s + (0.100 − 0.994i)4-s + (0.922 − 0.386i)5-s + (−0.867 + 0.497i)6-s + (0.120 − 0.992i)7-s + (0.592 + 0.805i)8-s + (0.909 + 0.416i)9-s + (−0.425 + 0.905i)10-s + (0.892 − 0.451i)11-s + (0.310 − 0.950i)12-s + (0.0552 + 0.998i)13-s + (0.576 + 0.816i)14-s + (0.983 − 0.181i)15-s + (−0.979 − 0.200i)16-s + (−0.945 − 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.972 - 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.062801788 - 0.3648701426i\)
\(L(\frac12)\) \(\approx\) \(3.062801788 - 0.3648701426i\)
\(L(1)\) \(\approx\) \(1.415216428 + 0.1130420160i\)
\(L(1)\) \(\approx\) \(1.415216428 + 0.1130420160i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.741 + 0.670i)T \)
3 \( 1 + (0.977 + 0.212i)T \)
5 \( 1 + (0.922 - 0.386i)T \)
7 \( 1 + (0.120 - 0.992i)T \)
11 \( 1 + (0.892 - 0.451i)T \)
13 \( 1 + (0.0552 + 0.998i)T \)
17 \( 1 + (-0.945 - 0.325i)T \)
19 \( 1 + (0.285 - 0.958i)T \)
23 \( 1 + (-0.389 + 0.921i)T \)
29 \( 1 + (0.638 - 0.769i)T \)
31 \( 1 + (0.966 - 0.257i)T \)
37 \( 1 + (0.383 - 0.923i)T \)
41 \( 1 + (-0.682 + 0.730i)T \)
43 \( 1 + (0.999 - 0.0325i)T \)
47 \( 1 + (0.807 - 0.589i)T \)
53 \( 1 + (-0.247 + 0.968i)T \)
59 \( 1 + (-0.197 + 0.980i)T \)
61 \( 1 + (-0.974 - 0.225i)T \)
67 \( 1 + (0.260 + 0.965i)T \)
71 \( 1 + (0.951 - 0.307i)T \)
73 \( 1 + (-0.247 - 0.968i)T \)
79 \( 1 + (-0.999 + 0.0260i)T \)
83 \( 1 + (0.929 + 0.368i)T \)
89 \( 1 + (0.705 - 0.708i)T \)
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

−21.45877875753849607910828586267, −20.52895185597987042620376187582, −20.152858777039119145996652019479, −19.10714948565409976872488630701, −18.55538918911077053622145439423, −17.821541034440667460207351732923, −17.32028859957897236253786155883, −16.013328068936007022036015779034, −15.18323327988513486279290947476, −14.36909718679605948731818118530, −13.56789505529953513412160272112, −12.53955534087020809687792700072, −12.22177544351807232487130026996, −10.865923705564095856947921243137, −10.06313353518480757174716968904, −9.42913450972543767112857126217, −8.63822734808860337227083383626, −8.07943021131467212262097983863, −6.86313721026843273051231135106, −6.17960713575216507158994757513, −4.67891852806921162187739425248, −3.48098486836103554967404107653, −2.649068590986535385587472471220, −1.99573146702204489808648471307, −1.14376416718270488763486471071, 0.78860914819309150096721826935, 1.61686865731687634142607119901, 2.577660386634071532421237715415, 4.15583248205253600866439967688, 4.71966597035634185248413964106, 6.10986382690737850838625500390, 6.84728890980398204995521247674, 7.62764951479576193523288000673, 8.71848003969653293907911494612, 9.2202648833075585075344946977, 9.80566394977146404170320630105, 10.72304630257903530489178385230, 11.65006570415836073721334121962, 13.34456764639766457489936481180, 13.84811877276374828012624056936, 14.14306694999792940379217138271, 15.30143005783116663969143539097, 16.066178621723409219339311633545, 16.85935841898020858268080012740, 17.455815020482405198291476845540, 18.259574942689352338465387254314, 19.38500365789972908580216304856, 19.76924168825322185636934205030, 20.51863694822144426395691962519, 21.39900496910396881101169137259

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