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Properties

Label 1-967-967.130-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.743 + 0.668i$
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  + (0.947 + 0.319i)2-s + (−0.638 − 0.769i)3-s + (0.795 + 0.605i)4-s + (−0.322 + 0.946i)5-s + (−0.359 − 0.933i)6-s + (−0.0162 + 0.999i)7-s + (0.560 + 0.828i)8-s + (−0.184 + 0.982i)9-s + (−0.608 + 0.793i)10-s + (−0.883 + 0.468i)11-s + (−0.0422 − 0.999i)12-s + (0.847 − 0.530i)13-s + (−0.334 + 0.942i)14-s + (0.934 − 0.356i)15-s + (0.266 + 0.963i)16-s + (−0.977 − 0.212i)17-s + ⋯
L(s)  = 1  + (0.947 + 0.319i)2-s + (−0.638 − 0.769i)3-s + (0.795 + 0.605i)4-s + (−0.322 + 0.946i)5-s + (−0.359 − 0.933i)6-s + (−0.0162 + 0.999i)7-s + (0.560 + 0.828i)8-s + (−0.184 + 0.982i)9-s + (−0.608 + 0.793i)10-s + (−0.883 + 0.468i)11-s + (−0.0422 − 0.999i)12-s + (0.847 − 0.530i)13-s + (−0.334 + 0.942i)14-s + (0.934 − 0.356i)15-s + (0.266 + 0.963i)16-s + (−0.977 − 0.212i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.743 + 0.668i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.743 + 0.668i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5427225892 + 1.416113756i\)
\(L(\frac12)\) \(\approx\) \(0.5427225892 + 1.416113756i\)
\(L(1)\) \(\approx\) \(1.162003731 + 0.5681379817i\)
\(L(1)\) \(\approx\) \(1.162003731 + 0.5681379817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.947 + 0.319i)T \)
3 \( 1 + (-0.638 - 0.769i)T \)
5 \( 1 + (-0.322 + 0.946i)T \)
7 \( 1 + (-0.0162 + 0.999i)T \)
11 \( 1 + (-0.883 + 0.468i)T \)
13 \( 1 + (0.847 - 0.530i)T \)
17 \( 1 + (-0.977 - 0.212i)T \)
19 \( 1 + (0.719 - 0.694i)T \)
23 \( 1 + (0.938 + 0.344i)T \)
29 \( 1 + (-0.260 + 0.965i)T \)
31 \( 1 + (-0.566 + 0.824i)T \)
37 \( 1 + (-0.533 + 0.845i)T \)
41 \( 1 + (-0.0682 - 0.997i)T \)
43 \( 1 + (-0.870 + 0.491i)T \)
47 \( 1 + (-0.996 - 0.0844i)T \)
53 \( 1 + (-0.877 - 0.480i)T \)
59 \( 1 + (0.581 - 0.813i)T \)
61 \( 1 + (0.898 + 0.439i)T \)
67 \( 1 + (0.996 - 0.0779i)T \)
71 \( 1 + (-0.750 - 0.660i)T \)
73 \( 1 + (-0.877 + 0.480i)T \)
79 \( 1 + (0.663 + 0.748i)T \)
83 \( 1 + (-0.953 - 0.300i)T \)
89 \( 1 + (-0.430 + 0.902i)T \)
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

−21.27521222445250290353938282496, −20.75560381136589347061368852202, −20.37462702888630519068754693670, −19.39171747019737754590760608251, −18.35454916618167251668276522910, −17.14199897220657760561628931202, −16.43964409494722226294097156628, −15.97515917678027129916174922018, −15.25186756162941465869740062697, −14.19136462935818194623882655732, −13.22527867770264181553444287293, −12.87950581381288306467870369939, −11.53729044580561828727230456259, −11.264217782369676107673602841070, −10.375362800986381275042533257041, −9.5534219539912881597730861165, −8.44768901523668085923186165065, −7.29806444870428058434026674281, −6.240152376817611314874113718915, −5.45409901374439514258192894227, −4.609967229056012498544845473580, −4.01642556021396115085838194348, −3.25481755056289649834907870001, −1.61426262050733904906443170191, −0.49218493028799988364954719596, 1.75826081534407165702268292051, 2.70411137040289637294242934257, 3.36422859613362082171498261970, 5.02078496351610748911524982408, 5.34385696509280432670713802380, 6.52795562678664253329456119512, 6.93873202888970513775616180969, 7.83932604922875624188561247031, 8.70907025721676935194149336518, 10.36909526780532928594799551923, 11.24642555181327139718504637782, 11.54284350949207923634529835271, 12.70078790167649950946543890124, 13.10537995352617753741296217362, 14.00857371176020604708370278854, 15.015005784902066764090241995383, 15.66437733452317214942076353526, 16.14104430619869377633598483406, 17.61789308375930963686116484244, 17.98627742350892683439867369005, 18.75096608354483824323173755117, 19.692072119733268279990001695229, 20.59945540876879950012833623280, 21.66605042671018717274896515611, 22.28352388372488572722090538652

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