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Properties

Label 1-712-712.219-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.232 + 0.972i$
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  + (0.0713 + 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

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

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.232 + 0.972i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.232 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.015286364 + 1.285965577i\)
\(L(\frac12)\) \(\approx\) \(1.015286364 + 1.285965577i\)
\(L(1)\) \(\approx\) \(1.086484501 + 0.5865177096i\)
\(L(1)\) \(\approx\) \(1.086484501 + 0.5865177096i\)

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 \( 1 + (-0.959 + 0.281i)T \)
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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

−22.42701204445912841198129501761, −21.43374270767610742950950976422, −20.734512031309300927567152880744, −19.81148509800583744934069257266, −19.18234890267729605981506524848, −18.15116904372723990817681086286, −17.66127760824294092129993453700, −16.78543559856554279663082697987, −15.89337204860213340174547677120, −14.81815746042608111908751236840, −13.989032653034253635115126233750, −13.0868527760383310635136892850, −12.60225667530581091897868227117, −11.52970332620077462903578272918, −11.09911982218249859560743925746, −9.194751442465951225758600015445, −8.86176837900194999186497276379, −8.18609956687845140028225802632, −6.85186107937486358047713303162, −6.13214799922198459201718080168, −5.241377904963763115653432502038, −4.18893650745046049698574397289, −2.70441297773121858571859857190, −1.77936299019901683966454603935, −0.84994773447726062062314560044, 1.45027367178452783831187030966, 2.73245594360992839879959366752, 3.99475955071824824435881249770, 4.17443535320386136717774559102, 5.75169367285877348625465686440, 6.46480522874338752738102999824, 7.55145949954604925570740171355, 8.5902575647822665781170177449, 9.537344644469585166850627770158, 10.34804022265283769199629083043, 11.0867708882701443604666710396, 11.52853129083555115098956865486, 13.19805036163997199529234341036, 13.92101126904114340511786642303, 14.77477004643776275038499483711, 15.18720791801704005328286301050, 16.31718710016524821259198296670, 17.25040963103422951753366559028, 17.61253929685926984474525419198, 18.87712486642124300027070790282, 19.66652812440080264871253990510, 20.61516062795218064932624003209, 21.140812059036326564101480798388, 22.04783353659228359466997843338, 22.81063734399397818748329582952

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