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Properties

Label 1-712-712.101-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.774 + 0.632i$
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  + (0.707 − 0.707i)3-s − i·5-s + (−0.707 + 0.707i)7-s − i·9-s + 11-s + (0.707 + 0.707i)13-s + (−0.707 − 0.707i)15-s + i·17-s + (−0.707 − 0.707i)19-s − i·21-s + (−0.707 + 0.707i)23-s − 25-s + (−0.707 − 0.707i)27-s + (−0.707 − 0.707i)29-s + (−0.707 − 0.707i)31-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s − i·5-s + (−0.707 + 0.707i)7-s − i·9-s + 11-s + (0.707 + 0.707i)13-s + (−0.707 − 0.707i)15-s + i·17-s + (−0.707 − 0.707i)19-s − i·21-s + (−0.707 + 0.707i)23-s − 25-s + (−0.707 − 0.707i)27-s + (−0.707 − 0.707i)29-s + (−0.707 − 0.707i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.774 + 0.632i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.774 + 0.632i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1322558526 - 0.3711662236i\)
\(L(\frac12)\) \(\approx\) \(-0.1322558526 - 0.3711662236i\)
\(L(1)\) \(\approx\) \(0.9844995361 - 0.3737712911i\)
\(L(1)\) \(\approx\) \(0.9844995361 - 0.3737712911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 - iT \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 - iT \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 - T \)
71 \( 1 + iT \)
73 \( 1 - T \)
79 \( 1 - iT \)
83 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + T \)
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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

−22.63873076724172343499327617929, −22.24444599785358267538777183915, −21.29053697317935176291275225239, −20.19928753241173182884855951307, −19.93057896334362458983882007008, −18.86345139853272783672370667823, −18.230116144228340617623647552592, −16.94084217393335448552397543083, −16.301387539305512924201494768049, −15.431777310608248858268003032929, −14.59181301177544022619958270185, −14.00402386653622728060251577554, −13.25769484428406373670139260128, −12.03493025747155195494221144877, −10.802922363282249726647702017534, −10.414546128994795452066799548877, −9.533452908424627486665280474561, −8.650796851837246044390803286538, −7.57130435167979189923216473530, −6.760153201788349208166133856938, −5.82545775505582187559404397622, −4.38037840074934212047549487878, −3.540607410706211209680820524668, −3.02400307724139983867131016967, −1.6710546063027783444224871803, 0.07504272837963578433343494883, 1.46499353457235547645742024911, 2.11879313972145279993738958340, 3.58626783027292779507946348677, 4.20938486986857186352481056452, 5.86518097948337826422974825156, 6.32106744315632198703945555656, 7.50186873957339557061002032549, 8.58791841106433843106223777668, 9.03633479773125051024753702186, 9.68809703488347084211843524723, 11.33964355894357082895282837652, 12.10911254163705323872882005803, 12.88738028920009949034370690243, 13.402225506425689538524088364388, 14.4251949892970354846122122557, 15.30540539290279994291758605582, 16.12251052440662003492616733255, 17.071406165212242967650934619851, 17.81391450116964183666665973129, 19.007842316849214992624352536325, 19.38378563035932510832310560577, 20.08558879906127691914170240604, 21.08271666678091609745055263633, 21.69938993624776689613107536250

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