Properties

Label 1-712-712.483-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: $\chi_{712} (483, \cdot )$
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

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

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