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Properties

Label 1-712-712.459-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.378 - 0.925i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (0.800 + 0.599i)3-s + (−0.540 − 0.841i)5-s + (0.212 − 0.977i)7-s + (0.281 + 0.959i)9-s + (−0.841 − 0.540i)11-s + (0.599 − 0.800i)13-s + (0.0713 − 0.997i)15-s + (−0.755 − 0.654i)17-s + (−0.877 − 0.479i)19-s + (0.755 − 0.654i)21-s + (−0.479 + 0.877i)23-s + (−0.415 + 0.909i)25-s + (−0.349 + 0.936i)27-s + (−0.977 − 0.212i)29-s + (−0.479 − 0.877i)31-s + ⋯
L(s)  = 1  + (0.800 + 0.599i)3-s + (−0.540 − 0.841i)5-s + (0.212 − 0.977i)7-s + (0.281 + 0.959i)9-s + (−0.841 − 0.540i)11-s + (0.599 − 0.800i)13-s + (0.0713 − 0.997i)15-s + (−0.755 − 0.654i)17-s + (−0.877 − 0.479i)19-s + (0.755 − 0.654i)21-s + (−0.479 + 0.877i)23-s + (−0.415 + 0.909i)25-s + (−0.349 + 0.936i)27-s + (−0.977 − 0.212i)29-s + (−0.479 − 0.877i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.378 - 0.925i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.378 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6114998944 - 0.9105358540i\)
\(L(\frac12)\) \(\approx\) \(0.6114998944 - 0.9105358540i\)
\(L(1)\) \(\approx\) \(1.023862565 - 0.2545029455i\)
\(L(1)\) \(\approx\) \(1.023862565 - 0.2545029455i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.800 + 0.599i)T \)
5 \( 1 + (-0.540 - 0.841i)T \)
7 \( 1 + (0.212 - 0.977i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (0.599 - 0.800i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (-0.877 - 0.479i)T \)
23 \( 1 + (-0.479 + 0.877i)T \)
29 \( 1 + (-0.977 - 0.212i)T \)
31 \( 1 + (-0.479 - 0.877i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.599 + 0.800i)T \)
43 \( 1 + (0.977 - 0.212i)T \)
47 \( 1 + (0.989 + 0.142i)T \)
53 \( 1 + (-0.989 + 0.142i)T \)
59 \( 1 + (-0.800 + 0.599i)T \)
61 \( 1 + (-0.936 - 0.349i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.540 - 0.841i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (0.281 - 0.959i)T \)
83 \( 1 + (0.0713 + 0.997i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−22.949030437018696477161986076382, −21.99375647712551907518277176712, −21.14362185377272279602833771568, −20.40785488115231627157731387064, −19.39675753061800511907553674158, −18.689559019829872239611321268598, −18.37923884332801117044836045631, −17.43599392801239001371528976953, −15.92561325716868741793439398821, −15.35298653505965840101440083697, −14.630726774744616727135364801970, −13.96555299920901969807715414067, −12.74296090936922278895829048737, −12.30812524310120879183806311093, −11.171424008243704914641426367295, −10.391423701094079945517782535926, −9.09370818111503753759639475294, −8.45420222283612988808179247976, −7.643289947044785309111895231596, −6.705000010011738098852097052131, −5.98003430551182426903441502393, −4.43951250001436981749886621571, −3.52747212342022382271210017763, −2.38250936596368302055485697818, −1.8898931917869344869994775536, 0.436999402659275575571309422557, 1.94186265160904601307780569290, 3.23393423603429712126518637554, 4.05260280766806438833661441806, 4.78522851464064973000360174835, 5.80292975427699565289543244731, 7.521140710503249037953424496089, 7.85579999444144770385898725578, 8.83797212339730407622592385263, 9.5567158590525825729879023371, 10.895051314352327281764500703124, 11.01495740031687596308212292551, 12.6810260273061145406450421814, 13.38745737510168188884308521507, 13.8933869517074421681346945966, 15.239409723080047718399095712826, 15.6675854965479002212074039573, 16.463159132425889974001971075138, 17.2679405406959289199818706342, 18.35907264321767457113288344309, 19.45266478217703651881693218020, 20.00760721142149179201627263730, 20.72477292239929452662800010563, 21.15399934527759547015897543220, 22.3153573741741588916408947766

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