Properties

Label 1-967-967.112-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.993 - 0.116i$
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  + (0.653 + 0.756i)2-s + (−0.981 + 0.193i)3-s + (−0.145 + 0.989i)4-s + (−0.737 − 0.675i)5-s + (−0.787 − 0.615i)6-s + (−0.833 + 0.552i)7-s + (−0.844 + 0.536i)8-s + (0.924 − 0.380i)9-s + (0.0292 − 0.999i)10-s + (0.279 − 0.960i)11-s + (−0.0487 − 0.998i)12-s + (−0.279 − 0.960i)13-s + (−0.962 − 0.269i)14-s + (0.854 + 0.519i)15-s + (−0.957 − 0.288i)16-s + (0.126 − 0.991i)17-s + ⋯
L(s)  = 1  + (0.653 + 0.756i)2-s + (−0.981 + 0.193i)3-s + (−0.145 + 0.989i)4-s + (−0.737 − 0.675i)5-s + (−0.787 − 0.615i)6-s + (−0.833 + 0.552i)7-s + (−0.844 + 0.536i)8-s + (0.924 − 0.380i)9-s + (0.0292 − 0.999i)10-s + (0.279 − 0.960i)11-s + (−0.0487 − 0.998i)12-s + (−0.279 − 0.960i)13-s + (−0.962 − 0.269i)14-s + (0.854 + 0.519i)15-s + (−0.957 − 0.288i)16-s + (0.126 − 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.993 - 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006222584133 + 0.1065880942i\)
\(L(\frac12)\) \(\approx\) \(0.006222584133 + 0.1065880942i\)
\(L(1)\) \(\approx\) \(0.7067459729 + 0.2157471529i\)
\(L(1)\) \(\approx\) \(0.7067459729 + 0.2157471529i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.653 - 0.756i)T \)
3 \( 1 + (0.981 - 0.193i)T \)
5 \( 1 + (0.737 + 0.675i)T \)
7 \( 1 + (0.833 - 0.552i)T \)
11 \( 1 + (-0.279 + 0.960i)T \)
13 \( 1 + (0.279 + 0.960i)T \)
17 \( 1 + (-0.126 + 0.991i)T \)
19 \( 1 + (-0.799 + 0.600i)T \)
23 \( 1 + (0.996 - 0.0779i)T \)
29 \( 1 + (-0.998 + 0.0585i)T \)
31 \( 1 + (0.945 + 0.325i)T \)
37 \( 1 + (0.165 - 0.986i)T \)
41 \( 1 + (-0.775 + 0.631i)T \)
43 \( 1 + (0.938 - 0.344i)T \)
47 \( 1 + (-0.995 - 0.0974i)T \)
53 \( 1 + (-0.682 - 0.730i)T \)
59 \( 1 + (0.932 - 0.362i)T \)
61 \( 1 + (0.775 - 0.631i)T \)
67 \( 1 + (-0.945 + 0.325i)T \)
71 \( 1 + (-0.653 + 0.756i)T \)
73 \( 1 + (-0.682 + 0.730i)T \)
79 \( 1 + (0.560 + 0.828i)T \)
83 \( 1 + (-0.00975 - 0.999i)T \)
89 \( 1 + (-0.945 + 0.325i)T \)
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

−21.564175397340494678160484646712, −20.08343165852744183233942647835, −19.75066725602999786623580741228, −18.84132960286905679469478097701, −18.27635942431689214365989482395, −17.28244810702313918223646279340, −16.230519148061797873830584738344, −15.68019595794886541317772967693, −14.58190457331944546927377865656, −13.96875771978343627233560041895, −12.79432411105683672861118267750, −12.2393736146687314191328180623, −11.694579325245462405199937735257, −10.70110781305075264014843006563, −10.18361580724480850732590556129, −9.42292442163953875873408341998, −7.71671979689052655513335554530, −6.79086609702624271378941496494, −6.36087919141082195115991792534, −5.21281500864460931186902900458, −4.01948375712260817520274807094, −3.842540454599911435004901386075, −2.34316885509492543435976353699, −1.31100799741656892927476042975, −0.0311420483963184190224438696, 0.74158689142797656296553117682, 2.89916928894000403123301557719, 3.65872564565486467063763270272, 4.69787222993813766297588205675, 5.45283345375596531830799668114, 6.03260583222291131105229901404, 7.04897717944225694112993354537, 7.85124971691178959517105747637, 8.88945013325804319707659922761, 9.65048866234166355868444309067, 10.9776712077508542123725950375, 12.03564652397498506179820870313, 12.11764538641507739922298175427, 13.1543491251109288170707465263, 13.87711509359678407138349811460, 15.3456249590287106322526448781, 15.62573757334526298145203088723, 16.394667809069920463481576254972, 16.79230731494853343475957106945, 17.90383418842154866303661905992, 18.539382666724482705558344780191, 19.73330143917133919947986569005, 20.50633381205270547679462411572, 21.613596545386656096536633478722, 22.14102297715600056879608849191

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