↑

Properties

Label 1-712-712.165-r1-0-0
Degree $1$
Conductor $712$
Sign $0.232 + 0.972i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0713 + 0.997i)3-s + (0.281 − 0.959i)5-s + (−0.479 − 0.877i)7-s + (−0.989 − 0.142i)9-s + (−0.959 + 0.281i)11-s + (0.997 + 0.0713i)13-s + (0.936 + 0.349i)15-s + (−0.909 − 0.415i)17-s + (0.599 + 0.800i)19-s + (0.909 − 0.415i)21-s + (−0.800 + 0.599i)23-s + (−0.841 − 0.540i)25-s + (0.212 − 0.977i)27-s + (−0.877 + 0.479i)29-s + (−0.800 − 0.599i)31-s + ⋯
L(s)  = 1  + (−0.0713 + 0.997i)3-s + (0.281 − 0.959i)5-s + (−0.479 − 0.877i)7-s + (−0.989 − 0.142i)9-s + (−0.959 + 0.281i)11-s + (0.997 + 0.0713i)13-s + (0.936 + 0.349i)15-s + (−0.909 − 0.415i)17-s + (0.599 + 0.800i)19-s + (0.909 − 0.415i)21-s + (−0.800 + 0.599i)23-s + (−0.841 − 0.540i)25-s + (0.212 − 0.977i)27-s + (−0.877 + 0.479i)29-s + (−0.800 − 0.599i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.232 + 0.972i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.232 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8988281013 + 0.7096363471i\)
\(L(\frac12)\) \(\approx\) \(0.8988281013 + 0.7096363471i\)
\(L(1)\) \(\approx\) \(0.8801918375 + 0.1152719537i\)
\(L(1)\) \(\approx\) \(0.8801918375 + 0.1152719537i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.0713 + 0.997i)T \)
5 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (-0.479 - 0.877i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (0.997 + 0.0713i)T \)
17 \( 1 + (-0.909 - 0.415i)T \)
19 \( 1 + (0.599 + 0.800i)T \)
23 \( 1 + (-0.800 + 0.599i)T \)
29 \( 1 + (-0.877 + 0.479i)T \)
31 \( 1 + (-0.800 - 0.599i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.997 - 0.0713i)T \)
43 \( 1 + (-0.877 - 0.479i)T \)
47 \( 1 + (0.755 + 0.654i)T \)
53 \( 1 + (0.755 - 0.654i)T \)
59 \( 1 + (0.0713 + 0.997i)T \)
61 \( 1 + (0.977 + 0.212i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (0.281 + 0.959i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (0.989 - 0.142i)T \)
83 \( 1 + (0.936 - 0.349i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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

−22.20647094823091902305280399578, −21.70566538968432155329629631109, −20.483076206497314330271678305280, −19.59349378207503974114094152712, −18.71007263894919511798790869572, −18.23150329046358150074142686776, −17.811647193061536532301374412400, −16.470698641612044924895026100144, −15.559325879003441403110950139652, −14.842735368217630346673058423629, −13.64059766209955821601976160822, −13.29312099727938855834741306378, −12.34622378122482908210878703747, −11.27178551972931012685937175579, −10.799428227035418705550508199237, −9.521247830582050276698509229698, −8.5537129631965983699574285217, −7.73285327872807997807069076272, −6.65056925498380967199085398232, −6.11651967509490100600488511610, −5.302979919453108048547237210153, −3.53561590528332631727098397398, −2.61889464565206996916860774329, −1.971908977037563722494802491098, −0.34501245673553009427214573123, 0.78353041519705978392853827270, 2.23396071532183493652302088487, 3.68623167316681861984950873360, 4.1891538784194849921489832624, 5.34358769403744491163234481570, 5.925331582373378962423193488242, 7.36043979004741488237132357853, 8.33038609871239509562473048746, 9.287280933863799441230694015, 9.88946436555390849558124625334, 10.75575523486266569583962026819, 11.558074524187964723912763120265, 12.82545796623250552808440473401, 13.43564101608415992485364662840, 14.250507876879028720608607714832, 15.49324225444101960768056822373, 16.18393794882141855549493965283, 16.51049994125076206534449973351, 17.57693093109407683227753400931, 18.29154238668623015326598707490, 19.707504221984960343289715062027, 20.52846368275149503159336929258, 20.61069195372404689953913923710, 21.68559547975346943277054097542, 22.54273422895712995570615584375

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