Properties

Label 1-967-967.115-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.755 + 0.655i$
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  + (0.771 − 0.636i)2-s + (−0.822 + 0.568i)3-s + (0.190 − 0.981i)4-s + (0.304 − 0.952i)5-s + (−0.272 + 0.962i)6-s + (−0.961 − 0.276i)7-s + (−0.477 − 0.878i)8-s + (0.353 − 0.935i)9-s + (−0.371 − 0.928i)10-s + (−0.511 − 0.859i)11-s + (0.401 + 0.915i)12-s + (−0.488 − 0.872i)13-s + (−0.917 + 0.398i)14-s + (0.291 + 0.956i)15-s + (−0.927 − 0.374i)16-s + (0.996 − 0.0779i)17-s + ⋯
L(s)  = 1  + (0.771 − 0.636i)2-s + (−0.822 + 0.568i)3-s + (0.190 − 0.981i)4-s + (0.304 − 0.952i)5-s + (−0.272 + 0.962i)6-s + (−0.961 − 0.276i)7-s + (−0.477 − 0.878i)8-s + (0.353 − 0.935i)9-s + (−0.371 − 0.928i)10-s + (−0.511 − 0.859i)11-s + (0.401 + 0.915i)12-s + (−0.488 − 0.872i)13-s + (−0.917 + 0.398i)14-s + (0.291 + 0.956i)15-s + (−0.927 − 0.374i)16-s + (0.996 − 0.0779i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.755 + 0.655i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.755 + 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2764691525 - 0.7409822942i\)
\(L(\frac12)\) \(\approx\) \(-0.2764691525 - 0.7409822942i\)
\(L(1)\) \(\approx\) \(0.7219082709 - 0.6176414195i\)
\(L(1)\) \(\approx\) \(0.7219082709 - 0.6176414195i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.771 - 0.636i)T \)
3 \( 1 + (-0.822 + 0.568i)T \)
5 \( 1 + (0.304 - 0.952i)T \)
7 \( 1 + (-0.961 - 0.276i)T \)
11 \( 1 + (-0.511 - 0.859i)T \)
13 \( 1 + (-0.488 - 0.872i)T \)
17 \( 1 + (0.996 - 0.0779i)T \)
19 \( 1 + (-0.999 + 0.00650i)T \)
23 \( 1 + (0.527 + 0.849i)T \)
29 \( 1 + (-0.724 - 0.689i)T \)
31 \( 1 + (0.602 + 0.797i)T \)
37 \( 1 + (0.0617 + 0.998i)T \)
41 \( 1 + (0.854 - 0.519i)T \)
43 \( 1 + (-0.783 + 0.620i)T \)
47 \( 1 + (-0.677 - 0.735i)T \)
53 \( 1 + (0.983 - 0.181i)T \)
59 \( 1 + (-0.807 - 0.589i)T \)
61 \( 1 + (0.0227 + 0.999i)T \)
67 \( 1 + (0.389 + 0.921i)T \)
71 \( 1 + (0.165 - 0.986i)T \)
73 \( 1 + (0.983 + 0.181i)T \)
79 \( 1 + (-0.395 + 0.918i)T \)
83 \( 1 + (-0.922 + 0.386i)T \)
89 \( 1 + (-0.992 - 0.123i)T \)
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

−22.56286187530284798895593028278, −21.64242788685465888768523849035, −21.176619853570791925546041844200, −19.75062070534325672277995065839, −18.7555016782781583440371509191, −18.375786954786409584259137088398, −17.23965903328926483709749484156, −16.783873096901869806690281074, −15.93205524199047889058825089281, −14.9829574689582096766932138304, −14.3880031532836777712648800158, −13.34552736239035158583213048444, −12.68630751211928569145820347757, −12.14502258559369817548459050613, −11.13887512039543553931577099906, −10.285865274618220634850308685012, −9.317430091579748793553705382348, −7.877879654107894367287808860213, −7.11221203485397850241027013474, −6.56429084746165957902337250859, −5.90112505666626589044307251944, −4.98959933772343614644991646716, −3.95905982398061968520068499521, −2.70932778865966788357396075360, −2.07029705513583375563935983899, 0.295206331851548223677966811138, 1.24262706161955856471388467119, 2.8489199002741153837903710740, 3.62720054019845550264353664767, 4.592067648413541023344906894033, 5.5201538351211074226515120441, 5.82265968660371569209029745845, 6.91968541185950249963235346073, 8.382492269849502557781700298, 9.60910052766575222380839247369, 9.98952258376645704273788273745, 10.79544586913344869234348459528, 11.73346450442720579987028220041, 12.56989359786019672790702647954, 13.02759184288430218360324685900, 13.80476438391416483974610286368, 15.08374290963546828133595435878, 15.67549002120222786585584988950, 16.55927838236636726075338998898, 17.02944918637053122325964062450, 18.16901511749753001873120379372, 19.19166624416783055923411986959, 19.81452374533585773027767904211, 20.85345142475722092980020307876, 21.25241084526514302953862627695

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