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Properties

Label 2-712-8.5-c1-0-35
Degree $2$
Conductor $712$
Sign $0.983 + 0.182i$
Analytic cond. $5.68534$
Root an. cond. $2.38439$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.780 − 1.17i)2-s + 0.662i·3-s + (−0.780 − 1.84i)4-s + 2.35i·5-s + (0.780 + 0.516i)6-s + (−2.78 − 0.516i)8-s + 2.56·9-s + (2.78 + 1.84i)10-s − 1.32i·11-s + (1.21 − 0.516i)12-s + 3.02i·13-s − 1.56·15-s + (−2.78 + 2.87i)16-s + 7.56·17-s + (2 − 3.02i)18-s + 1.98i·19-s + ⋯
L(s)  = 1  + (0.552 − 0.833i)2-s + 0.382i·3-s + (−0.390 − 0.920i)4-s + 1.05i·5-s + (0.318 + 0.211i)6-s + (−0.983 − 0.182i)8-s + 0.853·9-s + (0.879 + 0.582i)10-s − 0.399i·11-s + (0.351 − 0.149i)12-s + 0.837i·13-s − 0.403·15-s + (−0.695 + 0.718i)16-s + 1.83·17-s + (0.471 − 0.711i)18-s + 0.455i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.983 + 0.182i$
Analytic conductor: \(5.68534\)
Root analytic conductor: \(2.38439\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :1/2),\ 0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04669 - 0.188641i\)
\(L(\frac12)\) \(\approx\) \(2.04669 - 0.188641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.780 + 1.17i)T \)
89 \( 1 + T \)
good3 \( 1 - 0.662iT - 3T^{2} \)
5 \( 1 - 2.35iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 1.32iT - 11T^{2} \)
13 \( 1 - 3.02iT - 13T^{2} \)
17 \( 1 - 7.56T + 17T^{2} \)
19 \( 1 - 1.98iT - 19T^{2} \)
23 \( 1 + 1.56T + 23T^{2} \)
29 \( 1 - 3.02iT - 29T^{2} \)
31 \( 1 - 9.56T + 31T^{2} \)
37 \( 1 + 5.66iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 2.73iT - 43T^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 - 5.75iT - 53T^{2} \)
59 \( 1 + 6.41iT - 59T^{2} \)
61 \( 1 + 0.371iT - 61T^{2} \)
67 \( 1 + 0.743iT - 67T^{2} \)
71 \( 1 + 9.12T + 71T^{2} \)
73 \( 1 + 8.68T + 73T^{2} \)
79 \( 1 + 6.87T + 79T^{2} \)
83 \( 1 + 15.1iT - 83T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.25471244214269725271524215908, −10.06260864994948398691436605078, −9.033233143085274426878612266205, −7.72374813022437639602763070119, −6.68660198703521150017867563733, −5.83205459325486793598293150225, −4.68742639866148314173752736544, −3.70566477222574742304823993719, −2.93191714943825235177898372711, −1.47746079466093947890494921316, 1.09703461183321076611364682831, 2.99557350938751247404566916818, 4.31285241714829522905859417061, 5.02461290107210161330005037209, 5.94039017726221744748577209793, 6.94446363275862634419402994077, 7.928576480336082234724828293517, 8.262171990210358078361507612408, 9.551271666892631849779772768326, 10.14406939130325014873455906025

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