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Properties

Label 1-712-712.213-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.908 + 0.418i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + (0.212 + 0.977i)3-s + (−0.755 − 0.654i)5-s + (0.997 − 0.0713i)7-s + (−0.909 + 0.415i)9-s + (−0.654 − 0.755i)11-s + (0.977 − 0.212i)13-s + (0.479 − 0.877i)15-s + (−0.281 + 0.959i)17-s + (−0.936 − 0.349i)19-s + (0.281 + 0.959i)21-s + (0.349 − 0.936i)23-s + (0.142 + 0.989i)25-s + (−0.599 − 0.800i)27-s + (−0.0713 − 0.997i)29-s + (0.349 + 0.936i)31-s + ⋯
L(s)  = 1  + (0.212 + 0.977i)3-s + (−0.755 − 0.654i)5-s + (0.997 − 0.0713i)7-s + (−0.909 + 0.415i)9-s + (−0.654 − 0.755i)11-s + (0.977 − 0.212i)13-s + (0.479 − 0.877i)15-s + (−0.281 + 0.959i)17-s + (−0.936 − 0.349i)19-s + (0.281 + 0.959i)21-s + (0.349 − 0.936i)23-s + (0.142 + 0.989i)25-s + (−0.599 − 0.800i)27-s + (−0.0713 − 0.997i)29-s + (0.349 + 0.936i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.908 + 0.418i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.908 + 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1658598513 + 0.7555392202i\)
\(L(\frac12)\) \(\approx\) \(0.1658598513 + 0.7555392202i\)
\(L(1)\) \(\approx\) \(0.8986789009 + 0.2205318850i\)
\(L(1)\) \(\approx\) \(0.8986789009 + 0.2205318850i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.212 + 0.977i)T \)
5 \( 1 + (-0.755 - 0.654i)T \)
7 \( 1 + (0.997 - 0.0713i)T \)
11 \( 1 + (-0.654 - 0.755i)T \)
13 \( 1 + (0.977 - 0.212i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (-0.936 - 0.349i)T \)
23 \( 1 + (0.349 - 0.936i)T \)
29 \( 1 + (-0.0713 - 0.997i)T \)
31 \( 1 + (0.349 + 0.936i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.977 + 0.212i)T \)
43 \( 1 + (-0.0713 + 0.997i)T \)
47 \( 1 + (-0.540 - 0.841i)T \)
53 \( 1 + (-0.540 + 0.841i)T \)
59 \( 1 + (-0.212 + 0.977i)T \)
61 \( 1 + (0.800 - 0.599i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (-0.755 + 0.654i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (0.909 + 0.415i)T \)
83 \( 1 + (0.479 + 0.877i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−22.26145352766282160089526344907, −20.84998135348960253941208208795, −20.5765161343291530165708168760, −19.372985291894294364838436008313, −18.81808401670046848426426630788, −17.94983365118332977526456964867, −17.633012207809227909355816817271, −16.21010902215938113526933751780, −15.28479141169586747921071281884, −14.60971778592982399701156198318, −13.825068681027158485565190299604, −12.92214017751536151373304557687, −11.99442597701382930011961331091, −11.27465394770591645226473153267, −10.64396765887195339950054352011, −9.134638957368375879290940095974, −8.24543702544873089844740415959, −7.53802473904450337189031099583, −6.92769125805842467782346485027, −5.81219361419968161961992348906, −4.66415262055314406068054141773, −3.54352925305404502471529406798, −2.43773622793653144801579706305, −1.57409694051579200302289704674, −0.1825482299916544335948184348, 1.14276758935618219472273118806, 2.62666187149519491200121095689, 3.78945325288958743141259880584, 4.47269571766375038847687994606, 5.252653362320421021692455876511, 6.3117229870607616854625037060, 8.01066524927745428177146818611, 8.35236561810050570161234217404, 8.97775815302772596093756316571, 10.4977013018389831023747495266, 10.90100121404968943923747519832, 11.69083435955170659847031658131, 12.881602439784114258206506996215, 13.70784310963471007793243928598, 14.78666353425735904762244031619, 15.37863947257893518443614906981, 16.1090081839536697014629230226, 16.8755688772363838435048673413, 17.67400383433277301993979415396, 18.86557336831441599424568686485, 19.6258036731647130456179932708, 20.54248406970795682218697006977, 21.106756102856924106921391395, 21.57373523749737372128189464921, 22.875197557923694914942256089254

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