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Properties

Label 1-712-712.379-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.929 - 0.368i$
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  + (−0.599 − 0.800i)3-s + (0.540 − 0.841i)5-s + (0.977 − 0.212i)7-s + (−0.281 + 0.959i)9-s + (−0.841 + 0.540i)11-s + (−0.800 + 0.599i)13-s + (−0.997 + 0.0713i)15-s + (0.755 − 0.654i)17-s + (−0.479 − 0.877i)19-s + (−0.755 − 0.654i)21-s + (−0.877 + 0.479i)23-s + (−0.415 − 0.909i)25-s + (0.936 − 0.349i)27-s + (−0.212 − 0.977i)29-s + (−0.877 − 0.479i)31-s + ⋯
L(s)  = 1  + (−0.599 − 0.800i)3-s + (0.540 − 0.841i)5-s + (0.977 − 0.212i)7-s + (−0.281 + 0.959i)9-s + (−0.841 + 0.540i)11-s + (−0.800 + 0.599i)13-s + (−0.997 + 0.0713i)15-s + (0.755 − 0.654i)17-s + (−0.479 − 0.877i)19-s + (−0.755 − 0.654i)21-s + (−0.877 + 0.479i)23-s + (−0.415 − 0.909i)25-s + (0.936 − 0.349i)27-s + (−0.212 − 0.977i)29-s + (−0.877 − 0.479i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1595874662 - 0.8355717285i\)
\(L(\frac12)\) \(\approx\) \(0.1595874662 - 0.8355717285i\)
\(L(1)\) \(\approx\) \(0.7447026571 - 0.4277136248i\)
\(L(1)\) \(\approx\) \(0.7447026571 - 0.4277136248i\)

Euler product

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

−22.82519819394606334862137952227, −21.99095900608083583838500428848, −21.415306654236040529493175100184, −20.8975033299389359591649036626, −19.82912403558173935889931255562, −18.4511469037924290681526257627, −18.18665124692736681929076577080, −17.20342570425941471201324023450, −16.54956678648500375536443171046, −15.47757071749082459769810862137, −14.60615689272724020388375992630, −14.37610735182296300324413999661, −12.91281273904584748169916138028, −12.01048098411708015480799963634, −11.06180302615260033266186111711, −10.37156799447566721553344905725, −9.95595825419052007853080660879, −8.566250849846959539113808411525, −7.77478924946012867802980197062, −6.52433703603290461171325883073, −5.54930071022944509817763432129, −5.12498218399158179430306228979, −3.77731298292779896476235355826, −2.846104356624899744947079304864, −1.61999903012125148320121970374, 0.41479137020523317136418257949, 1.81927830685133347582774743444, 2.2741673069832629306131149338, 4.30547306744285155536192200749, 5.15576277255141540563766649254, 5.63530218815500073204176079128, 7.047846777677802696946482634101, 7.647507500789774062994087047283, 8.5602580858472536540487582810, 9.6842907849822697626952278088, 10.562494788272129794810190049415, 11.68393489176457409933479705254, 12.16135064876934621632262117103, 13.16530960053878168496591557782, 13.75557439693838911396621817310, 14.666096688488663208786385477450, 15.8844109429059288792381348255, 16.78596839689116720604159881648, 17.44304089152159662904838625576, 17.946237992781579548331296294586, 18.85632097137473861770975887515, 19.837805706160245732149291946846, 20.686591349526654995316467840862, 21.37344707159605126813970122973, 22.22896503255951275843531366913

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