↑

Properties

Label 1-916-916.155-r0-0-0
Degree $1$
Conductor $916$
Sign $0.764 - 0.645i$
Analytic cond. $4.25388$
Root an. cond. $4.25388$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.635 + 0.771i)3-s + (−0.851 − 0.523i)5-s + (0.805 + 0.592i)7-s + (−0.191 − 0.981i)9-s + (−0.0825 − 0.996i)11-s + (0.475 + 0.879i)13-s + (0.945 − 0.324i)15-s + (−0.879 + 0.475i)17-s + (−0.0275 − 0.999i)19-s + (−0.969 + 0.245i)21-s + (0.218 + 0.975i)23-s + (0.451 + 0.892i)25-s + (0.879 + 0.475i)27-s + (0.110 − 0.993i)29-s + (−0.569 − 0.821i)31-s + ⋯
L(s)  = 1  + (−0.635 + 0.771i)3-s + (−0.851 − 0.523i)5-s + (0.805 + 0.592i)7-s + (−0.191 − 0.981i)9-s + (−0.0825 − 0.996i)11-s + (0.475 + 0.879i)13-s + (0.945 − 0.324i)15-s + (−0.879 + 0.475i)17-s + (−0.0275 − 0.999i)19-s + (−0.969 + 0.245i)21-s + (0.218 + 0.975i)23-s + (0.451 + 0.892i)25-s + (0.879 + 0.475i)27-s + (0.110 − 0.993i)29-s + (−0.569 − 0.821i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(916\)    =    \(2^{2} \cdot 229\)
Sign: $0.764 - 0.645i$
Analytic conductor: \(4.25388\)
Root analytic conductor: \(4.25388\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{916} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 916,\ (0:\ ),\ 0.764 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7694035193 - 0.2813819827i\)
\(L(\frac12)\) \(\approx\) \(0.7694035193 - 0.2813819827i\)
\(L(1)\) \(\approx\) \(0.7653973333 + 0.03968613078i\)
\(L(1)\) \(\approx\) \(0.7653973333 + 0.03968613078i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.635 + 0.771i)T \)
5 \( 1 + (-0.851 - 0.523i)T \)
7 \( 1 + (0.805 + 0.592i)T \)
11 \( 1 + (-0.0825 - 0.996i)T \)
13 \( 1 + (0.475 + 0.879i)T \)
17 \( 1 + (-0.879 + 0.475i)T \)
19 \( 1 + (-0.0275 - 0.999i)T \)
23 \( 1 + (0.218 + 0.975i)T \)
29 \( 1 + (0.110 - 0.993i)T \)
31 \( 1 + (-0.569 - 0.821i)T \)
37 \( 1 + (-0.962 - 0.272i)T \)
41 \( 1 + (0.656 - 0.754i)T \)
43 \( 1 + (-0.245 - 0.969i)T \)
47 \( 1 + (-0.981 + 0.191i)T \)
53 \( 1 + (-0.986 - 0.164i)T \)
59 \( 1 + (-0.272 - 0.962i)T \)
61 \( 1 + (0.945 + 0.324i)T \)
67 \( 1 + (0.656 + 0.754i)T \)
71 \( 1 + (0.904 - 0.426i)T \)
73 \( 1 + (0.892 - 0.451i)T \)
79 \( 1 + (-0.110 - 0.993i)T \)
83 \( 1 + (-0.716 - 0.697i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.998 - 0.0550i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.417266463635742796226186704025, −21.12035254276714826817108773863, −20.09714901247824036757966900320, −19.81736722948143671840586606220, −18.44011580283418672217779074600, −18.19957652844036347187862547343, −17.4411970362525791968272112905, −16.47168383969048595504599370560, −15.681922726708443100947619855092, −14.67330938468911426895206095336, −14.09433823381461865335554710728, −12.86807472130782276705447363105, −12.40222344506832529227959323459, −11.35488028512577011696839007451, −10.85838374738842041371566470962, −10.13835797263370734454188404055, −8.47042410784128449158347214617, −7.887667275390265597327101760780, −7.08264378105179333785034989436, −6.49973643913096028816569816923, −5.14067090736345125382126936273, −4.50363021379076047456064902398, −3.3013506337093744646137749997, −2.0784000894038656430776173859, −1.02340856634975732555937081003, 0.47850198095922524122972901642, 1.90768415945075793251886187167, 3.43346263649485126532924620092, 4.19324343089899057575180473806, 4.99298864822083171859310051141, 5.77260604202956427317398022311, 6.78437015727619479135396597891, 8.02999644806114398308991299303, 8.84647804784333616814013833774, 9.304442613565764915929156806870, 10.82539837277813702201389735939, 11.36418055629710963885923323083, 11.70972471000986723851554621721, 12.8424511734803179657845393698, 13.835292089822839945115742512897, 14.91374459546414203219545630554, 15.67336598762767407750517265579, 15.99436151211035241567905309649, 17.10134341702707671292299865584, 17.58134351139963388220746488238, 18.740337141472461498802925468197, 19.3748244404416848645549331001, 20.4420412140293858700405980425, 21.15720004450578544896889837496, 21.701386542632457769382880332856

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