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Properties

Label 1-967-967.2-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.692 + 0.721i$
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  + (0.304 − 0.952i)2-s + (0.0487 − 0.998i)3-s + (−0.815 − 0.579i)4-s + (−0.759 + 0.651i)5-s + (−0.936 − 0.350i)6-s + (0.929 − 0.368i)7-s + (−0.799 + 0.600i)8-s + (−0.995 − 0.0974i)9-s + (0.389 + 0.921i)10-s + (0.316 − 0.948i)11-s + (−0.618 + 0.785i)12-s + (0.663 − 0.748i)13-s + (−0.0682 − 0.997i)14-s + (0.613 + 0.789i)15-s + (0.328 + 0.944i)16-s + (0.353 − 0.935i)17-s + ⋯
L(s)  = 1  + (0.304 − 0.952i)2-s + (0.0487 − 0.998i)3-s + (−0.815 − 0.579i)4-s + (−0.759 + 0.651i)5-s + (−0.936 − 0.350i)6-s + (0.929 − 0.368i)7-s + (−0.799 + 0.600i)8-s + (−0.995 − 0.0974i)9-s + (0.389 + 0.921i)10-s + (0.316 − 0.948i)11-s + (−0.618 + 0.785i)12-s + (0.663 − 0.748i)13-s + (−0.0682 − 0.997i)14-s + (0.613 + 0.789i)15-s + (0.328 + 0.944i)16-s + (0.353 − 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.692 + 0.721i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.692 + 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4334308566 - 1.017265834i\)
\(L(\frac12)\) \(\approx\) \(-0.4334308566 - 1.017265834i\)
\(L(1)\) \(\approx\) \(0.5413214510 - 0.8429267420i\)
\(L(1)\) \(\approx\) \(0.5413214510 - 0.8429267420i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.304 - 0.952i)T \)
3 \( 1 + (0.0487 - 0.998i)T \)
5 \( 1 + (-0.759 + 0.651i)T \)
7 \( 1 + (0.929 - 0.368i)T \)
11 \( 1 + (0.316 - 0.948i)T \)
13 \( 1 + (0.663 - 0.748i)T \)
17 \( 1 + (0.353 - 0.935i)T \)
19 \( 1 + (0.100 + 0.994i)T \)
23 \( 1 + (-0.999 - 0.0195i)T \)
29 \( 1 + (-0.696 - 0.717i)T \)
31 \( 1 + (0.177 - 0.984i)T \)
37 \( 1 + (-0.171 - 0.985i)T \)
41 \( 1 + (-0.576 - 0.816i)T \)
43 \( 1 + (0.818 + 0.574i)T \)
47 \( 1 + (-0.235 + 0.971i)T \)
53 \( 1 + (-0.949 - 0.313i)T \)
59 \( 1 + (-0.347 + 0.937i)T \)
61 \( 1 + (-0.419 + 0.907i)T \)
67 \( 1 + (0.763 - 0.646i)T \)
71 \( 1 + (-0.977 - 0.212i)T \)
73 \( 1 + (-0.949 + 0.313i)T \)
79 \( 1 + (0.719 - 0.694i)T \)
83 \( 1 + (-0.132 + 0.991i)T \)
89 \( 1 + (-0.941 - 0.337i)T \)
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

−22.05249963965862403604599741863, −21.73249464981938010730637911978, −20.72622454166075788201304438731, −20.19629603771421945368372378379, −19.08219416044004947662569023481, −17.98347981061138296068806568758, −17.24306446555051888171980938662, −16.598177062731679857924080974772, −15.74095434788433529891554998101, −15.27296359314949543136894319266, −14.5744523943541943831562325072, −13.832017263142528908496308726437, −12.62597908547472699369442302300, −11.880396729715158859646829772800, −11.13895949723403134416142289743, −9.87737628898991476480070360993, −8.912887059765845148131443869784, −8.503572678767511640150502943319, −7.6641648660460275721927792273, −6.55493342806747851986278004154, −5.429743852493277678626716802930, −4.71671928426265450109085802184, −4.19529585487115225365596978417, −3.333205446298978273058650668873, −1.62021754439028342278254124653, 0.45675076996595053794299191761, 1.42299335839390677162340771313, 2.52380460456963982226633418500, 3.436001806085480867468222833142, 4.15601826828593437327381879243, 5.582056470535949318977197670731, 6.16046448472515943115221892479, 7.65575605340469461491129441794, 7.97013764043806428878125468386, 8.96121871463025042927740681272, 10.286357965622672483825828517531, 11.12577311118449583991947498870, 11.567514027618000335872545891547, 12.24285260728069598807928198526, 13.27297825952407165193442170268, 14.15810211001234319144177690396, 14.32569020095821687936290806083, 15.48215638370649034282415675205, 16.69157404737191965792605348483, 17.828414715596031796206344037245, 18.2982437546801364144803199623, 18.96721695113884973472899604742, 19.59197090741986240699194510665, 20.54926784610622025858169124770, 20.89753202099650407183829637220

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