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Properties

Label 1-712-712.171-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.732 + 0.680i$
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  + (−0.877 + 0.479i)3-s + (0.909 − 0.415i)5-s + (−0.936 − 0.349i)7-s + (0.540 − 0.841i)9-s + (−0.415 + 0.909i)11-s + (0.479 + 0.877i)13-s + (−0.599 + 0.800i)15-s + (−0.989 − 0.142i)17-s + (−0.212 − 0.977i)19-s + (0.989 − 0.142i)21-s + (−0.977 + 0.212i)23-s + (0.654 − 0.755i)25-s + (−0.0713 + 0.997i)27-s + (−0.349 + 0.936i)29-s + (−0.977 − 0.212i)31-s + ⋯
L(s)  = 1  + (−0.877 + 0.479i)3-s + (0.909 − 0.415i)5-s + (−0.936 − 0.349i)7-s + (0.540 − 0.841i)9-s + (−0.415 + 0.909i)11-s + (0.479 + 0.877i)13-s + (−0.599 + 0.800i)15-s + (−0.989 − 0.142i)17-s + (−0.212 − 0.977i)19-s + (0.989 − 0.142i)21-s + (−0.977 + 0.212i)23-s + (0.654 − 0.755i)25-s + (−0.0713 + 0.997i)27-s + (−0.349 + 0.936i)29-s + (−0.977 − 0.212i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.732 + 0.680i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.732 + 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1722486088 + 0.4386072434i\)
\(L(\frac12)\) \(\approx\) \(0.1722486088 + 0.4386072434i\)
\(L(1)\) \(\approx\) \(0.6778271857 + 0.1405916671i\)
\(L(1)\) \(\approx\) \(0.6778271857 + 0.1405916671i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.877 + 0.479i)T \)
5 \( 1 + (0.909 - 0.415i)T \)
7 \( 1 + (-0.936 - 0.349i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (0.479 + 0.877i)T \)
17 \( 1 + (-0.989 - 0.142i)T \)
19 \( 1 + (-0.212 - 0.977i)T \)
23 \( 1 + (-0.977 + 0.212i)T \)
29 \( 1 + (-0.349 + 0.936i)T \)
31 \( 1 + (-0.977 - 0.212i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.479 - 0.877i)T \)
43 \( 1 + (0.349 + 0.936i)T \)
47 \( 1 + (0.281 + 0.959i)T \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 + (0.877 + 0.479i)T \)
61 \( 1 + (-0.997 - 0.0713i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.909 - 0.415i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (0.540 + 0.841i)T \)
83 \( 1 + (-0.599 - 0.800i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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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

−22.24932558418041995782450117284, −21.816398726143284338393552582716, −20.83335046150555170394174369540, −19.66886676239617516537488288779, −18.76144276514590961853296415594, −18.22864436289651273116026933261, −17.56610726571299769028099827945, −16.50613933375965538561312543901, −16.0279327277220901813024959082, −14.92902422330681416007043172966, −13.67789732728007949409318905259, −13.17320914823104533992194806816, −12.49974477740696824032700164845, −11.342697460748445290711899105546, −10.55878771016846962652254954941, −9.95519023081298753066881012065, −8.781721401614682350741457777395, −7.72525519651093524003869774913, −6.584885357638407550384202456237, −5.878504773298061421279600338857, −5.5830114123686033839918757198, −3.94137217193698402022225049801, −2.741771301672672663721322824468, −1.80372681919358566546092134332, −0.24798597994207064039230686939, 1.38057799931392561153306143390, 2.58682325978174713222993722381, 4.08866902949222143678330804585, 4.702054968929585128804772780202, 5.82181303641528038080678500491, 6.50097727382572136103029728476, 7.29373268716660322841271848895, 9.134711900974541146835292674843, 9.36355230487244054989854025818, 10.38063971935846659697952035686, 11.05065817342667247322878810727, 12.2080940413133666195783163035, 12.97651783494403206323501978866, 13.582311988603481506155231609, 14.793192918829690869844462432335, 15.90455867752194976791976608604, 16.27456508561358748480387412627, 17.23925451763024645170450094667, 17.827903631610523149616493724648, 18.58428292579723568445733805307, 19.932225962629285765560221347851, 20.49627104700059355106639626951, 21.49191530459926783488298632976, 22.08255787875138811551000777804, 22.73186820078783528590912960352

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