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Properties

Label 1-967-967.430-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.292 + 0.956i$
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  + (−0.209 + 0.977i)2-s + (−0.977 + 0.212i)3-s + (−0.911 − 0.410i)4-s + (0.795 − 0.605i)5-s + (−0.00325 − 0.999i)6-s + (0.919 + 0.392i)7-s + (0.592 − 0.805i)8-s + (0.909 − 0.416i)9-s + (0.425 + 0.905i)10-s + (0.892 + 0.451i)11-s + (0.978 + 0.206i)12-s + (−0.837 − 0.547i)13-s + (−0.576 + 0.816i)14-s + (−0.648 + 0.761i)15-s + (0.663 + 0.748i)16-s + (−0.945 + 0.325i)17-s + ⋯
L(s)  = 1  + (−0.209 + 0.977i)2-s + (−0.977 + 0.212i)3-s + (−0.911 − 0.410i)4-s + (0.795 − 0.605i)5-s + (−0.00325 − 0.999i)6-s + (0.919 + 0.392i)7-s + (0.592 − 0.805i)8-s + (0.909 − 0.416i)9-s + (0.425 + 0.905i)10-s + (0.892 + 0.451i)11-s + (0.978 + 0.206i)12-s + (−0.837 − 0.547i)13-s + (−0.576 + 0.816i)14-s + (−0.648 + 0.761i)15-s + (0.663 + 0.748i)16-s + (−0.945 + 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.292 + 0.956i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (430, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6181514667 + 0.8358838372i\)
\(L(\frac12)\) \(\approx\) \(0.6181514667 + 0.8358838372i\)
\(L(1)\) \(\approx\) \(0.7197428053 + 0.4207992031i\)
\(L(1)\) \(\approx\) \(0.7197428053 + 0.4207992031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.209 + 0.977i)T \)
3 \( 1 + (-0.977 + 0.212i)T \)
5 \( 1 + (0.795 - 0.605i)T \)
7 \( 1 + (0.919 + 0.392i)T \)
11 \( 1 + (0.892 + 0.451i)T \)
13 \( 1 + (-0.837 - 0.547i)T \)
17 \( 1 + (-0.945 + 0.325i)T \)
19 \( 1 + (-0.687 + 0.726i)T \)
23 \( 1 + (0.389 + 0.921i)T \)
29 \( 1 + (-0.638 - 0.769i)T \)
31 \( 1 + (-0.705 + 0.708i)T \)
37 \( 1 + (0.991 + 0.129i)T \)
41 \( 1 + (0.682 + 0.730i)T \)
43 \( 1 + (0.471 + 0.881i)T \)
47 \( 1 + (0.914 - 0.404i)T \)
53 \( 1 + (-0.715 + 0.699i)T \)
59 \( 1 + (0.947 + 0.319i)T \)
61 \( 1 + (0.291 - 0.956i)T \)
67 \( 1 + (-0.260 + 0.965i)T \)
71 \( 1 + (0.951 + 0.307i)T \)
73 \( 1 + (-0.715 - 0.699i)T \)
79 \( 1 + (-0.522 + 0.852i)T \)
83 \( 1 + (-0.783 - 0.620i)T \)
89 \( 1 + (0.966 - 0.257i)T \)
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

−21.66392898927790612616993498495, −20.88401698719491760818146594830, −19.93151175213132812511999366073, −18.98233625474269917187439084993, −18.40938618742017117897535438482, −17.48867433296381535033046301691, −17.25109314167358262460808107337, −16.4790466157193558720769155315, −14.83287072739237794835616008481, −14.22037474716698992751358068736, −13.361302134026645495198143827947, −12.60965819197340787356730691357, −11.56471845219426899374655478055, −11.049628946758688286002756649439, −10.58982706167932549249881103479, −9.47884725342574725760962101107, −8.827596503904362548108576459935, −7.40155637063631088759020578417, −6.768004851398842398913867383491, −5.63112377697133051767846686356, −4.68581415492118002521856618243, −4.027067762849431814334197388352, −2.42208151562600398210663368734, −1.84368770801056590426052632620, −0.65747225459338642312523467081, 1.09571177330118915864734875962, 1.97022624129219792984534702304, 4.16837697472857929179777873525, 4.70035903979714057722281777291, 5.58565931218460332901326337343, 6.09093319854265893860441142387, 7.09926557908405903372484234776, 8.032673555079519128215067734871, 9.10339482268618543058795585416, 9.62361094165703640436360523510, 10.55070566991186136973177101227, 11.5302686822713473659613837564, 12.585509109274395569715387756761, 13.09244967632219618802136361276, 14.37489525142509177889397636124, 14.91249282062386674253207447552, 15.7252259717735557539238402166, 16.7538995586755751482801618939, 17.29706545785919803633768485740, 17.63165043670170073315571076653, 18.373198466694319056706063322115, 19.49189378154469666739952453709, 20.520929970053966636400005028844, 21.69279848932814356726929627926, 21.86346993182019847715905870717

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