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Properties

Label 1-967-967.720-r1-0-0
Degree $1$
Conductor $967$
Sign $0.887 + 0.460i$
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  + (0.811 + 0.584i)2-s + (−0.909 − 0.416i)3-s + (0.316 + 0.948i)4-s + (0.967 + 0.250i)5-s + (−0.494 − 0.869i)6-s + (−0.279 − 0.960i)7-s + (−0.297 + 0.954i)8-s + (0.653 + 0.756i)9-s + (0.638 + 0.769i)10-s + (0.592 − 0.805i)11-s + (0.107 − 0.994i)12-s + (−0.592 − 0.805i)13-s + (0.334 − 0.942i)14-s + (−0.775 − 0.631i)15-s + (−0.799 + 0.600i)16-s + (0.787 + 0.615i)17-s + ⋯
L(s)  = 1  + (0.811 + 0.584i)2-s + (−0.909 − 0.416i)3-s + (0.316 + 0.948i)4-s + (0.967 + 0.250i)5-s + (−0.494 − 0.869i)6-s + (−0.279 − 0.960i)7-s + (−0.297 + 0.954i)8-s + (0.653 + 0.756i)9-s + (0.638 + 0.769i)10-s + (0.592 − 0.805i)11-s + (0.107 − 0.994i)12-s + (−0.592 − 0.805i)13-s + (0.334 − 0.942i)14-s + (−0.775 − 0.631i)15-s + (−0.799 + 0.600i)16-s + (0.787 + 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (720, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.887 + 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.035642201 + 0.7401390503i\)
\(L(\frac12)\) \(\approx\) \(3.035642201 + 0.7401390503i\)
\(L(1)\) \(\approx\) \(1.509296936 + 0.3227376137i\)
\(L(1)\) \(\approx\) \(1.509296936 + 0.3227376137i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.811 + 0.584i)T \)
3 \( 1 + (-0.909 - 0.416i)T \)
5 \( 1 + (0.967 + 0.250i)T \)
7 \( 1 + (-0.279 - 0.960i)T \)
11 \( 1 + (0.592 - 0.805i)T \)
13 \( 1 + (-0.592 - 0.805i)T \)
17 \( 1 + (0.787 + 0.615i)T \)
19 \( 1 + (-0.892 + 0.451i)T \)
23 \( 1 + (0.696 + 0.717i)T \)
29 \( 1 + (0.184 + 0.982i)T \)
31 \( 1 + (-0.864 - 0.502i)T \)
37 \( 1 + (0.260 - 0.965i)T \)
41 \( 1 + (0.0682 + 0.997i)T \)
43 \( 1 + (0.442 - 0.896i)T \)
47 \( 1 + (0.977 - 0.212i)T \)
53 \( 1 + (0.854 - 0.519i)T \)
59 \( 1 + (0.126 + 0.991i)T \)
61 \( 1 + (-0.0682 - 0.997i)T \)
67 \( 1 + (0.864 - 0.502i)T \)
71 \( 1 + (0.811 - 0.584i)T \)
73 \( 1 + (0.854 + 0.519i)T \)
79 \( 1 + (0.544 + 0.838i)T \)
83 \( 1 + (-0.957 + 0.288i)T \)
89 \( 1 + (0.864 - 0.502i)T \)
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

−21.49919353638867874618545865673, −21.1069704577127374045848430285, −20.23302272370061165966862479024, −19.056540095287946461627831775420, −18.494627805828846288794662488163, −17.46835029505660855146433752901, −16.7763192982967301498101556583, −15.90821041096006038916419200429, −14.94339029931390525764570319335, −14.45714347472204902007366838407, −13.30903214417009329986317547872, −12.36891750006558861813067575258, −12.16258659802895491993746504239, −11.16316817305411090938866432700, −10.21946160057524037532785570717, −9.503306402935474944623465780821, −9.06120169882037437633590013472, −6.938907918720162504142077278157, −6.40525398804724485482010149834, −5.521159646552150547354507891728, −4.87705285492379024429225106731, −4.133508050841829267975547723974, −2.70651302007363628526099071913, −1.94363334317684031259352799397, −0.79464622202152751438195655932, 0.75578984929596688243388813545, 1.931428673572676501038683634664, 3.22998425289714351303747411758, 4.11644180782345421504202144790, 5.40022762601196891137154654986, 5.75261627837481852411938374352, 6.68683887833894668480624266720, 7.25960062562990511499398780107, 8.22713320763778583893493704989, 9.56676808838976427144042672223, 10.659975384893615020469040669048, 11.05083795087089264208189111545, 12.4038157706821985990884865397, 12.83732520983582601547866461071, 13.63198482354721412664700110979, 14.31378847098767173830170746703, 15.130230338251118235298367617143, 16.45567374626087125811784229686, 16.85767090257539350008825969695, 17.31107495035967719314985917677, 18.17481002865901218344038352887, 19.21861721703517208803873515536, 20.14181290103154730426562412485, 21.372283593233666570078049075715, 21.66978139618407867161641422239

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