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Properties

Label 1-967-967.4-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.336 + 0.941i$
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  + (−0.815 − 0.579i)2-s + (−0.995 − 0.0974i)3-s + (0.328 + 0.944i)4-s + (0.152 − 0.988i)5-s + (0.754 + 0.655i)6-s + (0.728 − 0.684i)7-s + (0.279 − 0.960i)8-s + (0.981 + 0.193i)9-s + (−0.696 + 0.717i)10-s + (−0.799 − 0.600i)11-s + (−0.235 − 0.971i)12-s + (−0.120 − 0.992i)13-s + (−0.990 + 0.136i)14-s + (−0.247 + 0.968i)15-s + (−0.783 + 0.620i)16-s + (−0.750 − 0.660i)17-s + ⋯
L(s)  = 1  + (−0.815 − 0.579i)2-s + (−0.995 − 0.0974i)3-s + (0.328 + 0.944i)4-s + (0.152 − 0.988i)5-s + (0.754 + 0.655i)6-s + (0.728 − 0.684i)7-s + (0.279 − 0.960i)8-s + (0.981 + 0.193i)9-s + (−0.696 + 0.717i)10-s + (−0.799 − 0.600i)11-s + (−0.235 − 0.971i)12-s + (−0.120 − 0.992i)13-s + (−0.990 + 0.136i)14-s + (−0.247 + 0.968i)15-s + (−0.783 + 0.620i)16-s + (−0.750 − 0.660i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.336 + 0.941i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1342413980 - 0.1906060195i\)
\(L(\frac12)\) \(\approx\) \(-0.1342413980 - 0.1906060195i\)
\(L(1)\) \(\approx\) \(0.3716814889 - 0.2921698019i\)
\(L(1)\) \(\approx\) \(0.3716814889 - 0.2921698019i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.815 - 0.579i)T \)
3 \( 1 + (-0.995 - 0.0974i)T \)
5 \( 1 + (0.152 - 0.988i)T \)
7 \( 1 + (0.728 - 0.684i)T \)
11 \( 1 + (-0.799 - 0.600i)T \)
13 \( 1 + (-0.120 - 0.992i)T \)
17 \( 1 + (-0.750 - 0.660i)T \)
19 \( 1 + (-0.979 + 0.200i)T \)
23 \( 1 + (0.999 + 0.0390i)T \)
29 \( 1 + (-0.0292 + 0.999i)T \)
31 \( 1 + (-0.936 - 0.350i)T \)
37 \( 1 + (-0.941 + 0.337i)T \)
41 \( 1 + (-0.334 + 0.942i)T \)
43 \( 1 + (0.341 + 0.940i)T \)
47 \( 1 + (-0.889 - 0.457i)T \)
53 \( 1 + (0.803 + 0.595i)T \)
59 \( 1 + (-0.759 - 0.651i)T \)
61 \( 1 + (-0.648 - 0.761i)T \)
67 \( 1 + (0.165 - 0.986i)T \)
71 \( 1 + (0.909 + 0.416i)T \)
73 \( 1 + (0.803 - 0.595i)T \)
79 \( 1 + (0.0357 - 0.999i)T \)
83 \( 1 + (-0.964 - 0.263i)T \)
89 \( 1 + (0.771 + 0.636i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−22.47592896186291002102616557290, −21.378632607816719934032810749761, −21.0596002522107061398253929063, −19.52697567928137796409581597328, −18.754176642158137880977343779036, −18.33243084757721800055883256222, −17.429813619506984287226636257417, −17.1841526920974663537851146105, −15.94184742136331905764302895853, −15.21227658350513148235628760802, −14.86548828508853530104060067397, −13.71525546032313365407208346278, −12.51914854634168793862761281901, −11.48909861561216479612022505244, −10.887812557355548285299818768222, −10.364326425220603876775758019604, −9.34633790303353208479295446995, −8.47545212181090656906166997458, −7.27765868818685433982218648251, −6.827268558334096169290965114869, −5.916602599335649605529876950573, −5.15703113490924897817402174112, −4.21843407500396317042998006852, −2.2962614123385632956422716154, −1.785720930932828760921137322955, 0.167563922429567619357854505412, 1.070548065489067626105755702596, 2.00111136259908858450099658262, 3.41877069669412604702885447996, 4.67867437704639825970180660566, 5.15454582394248039509044257457, 6.42990944098440918567785265550, 7.48547055560846254426146499250, 8.14467658527391782256722423756, 9.022082175217135794800836349733, 10.07617718522581458047958362807, 10.874100429544950953806364011300, 11.20906146932092890469248203164, 12.35400430432220576513688305734, 12.977127232535537032738660212890, 13.525093364501406827039488826972, 15.189008546311781782429726754105, 16.08705446033048744065009304048, 16.76273013916989576016302019579, 17.275067895545690446110340554641, 18.01786807749838215845164093576, 18.58425458519194269299819185184, 19.78206661165961229499399580033, 20.37274696212134783963743686178, 21.211385379700502490494926937761

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