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Properties

Label 2-1710-95.18-c1-0-21
Degree $2$
Conductor $1710$
Sign $0.136 + 0.990i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−2.23 − 0.0328i)5-s + (−3.17 + 3.17i)7-s + (0.707 − 0.707i)8-s + (1.55 + 1.60i)10-s − 1.49·11-s + (−2.38 + 2.38i)13-s + 4.49·14-s − 1.00·16-s + (−0.853 + 0.853i)17-s + (−4.34 − 0.349i)19-s + (0.0328 − 2.23i)20-s + (1.05 + 1.05i)22-s + (−4.67 − 4.67i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.999 − 0.0146i)5-s + (−1.20 + 1.20i)7-s + (0.250 − 0.250i)8-s + (0.492 + 0.507i)10-s − 0.451·11-s + (−0.662 + 0.662i)13-s + 1.20·14-s − 0.250·16-s + (−0.206 + 0.206i)17-s + (−0.996 − 0.0802i)19-s + (0.00733 − 0.499i)20-s + (0.225 + 0.225i)22-s + (−0.974 − 0.974i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.136 + 0.990i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.136 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3248396662\)
\(L(\frac12)\) \(\approx\) \(0.3248396662\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (2.23 + 0.0328i)T \)
19 \( 1 + (4.34 + 0.349i)T \)
good7 \( 1 + (3.17 - 3.17i)T - 7iT^{2} \)
11 \( 1 + 1.49T + 11T^{2} \)
13 \( 1 + (2.38 - 2.38i)T - 13iT^{2} \)
17 \( 1 + (0.853 - 0.853i)T - 17iT^{2} \)
23 \( 1 + (4.67 + 4.67i)T + 23iT^{2} \)
29 \( 1 - 5.40T + 29T^{2} \)
31 \( 1 + 0.364iT - 31T^{2} \)
37 \( 1 + (-4.29 - 4.29i)T + 37iT^{2} \)
41 \( 1 - 1.79iT - 41T^{2} \)
43 \( 1 + (-0.623 - 0.623i)T + 43iT^{2} \)
47 \( 1 + (-5.24 + 5.24i)T - 47iT^{2} \)
53 \( 1 + (5.27 - 5.27i)T - 53iT^{2} \)
59 \( 1 - 1.49T + 59T^{2} \)
61 \( 1 - 15.2T + 61T^{2} \)
67 \( 1 + (-0.989 - 0.989i)T + 67iT^{2} \)
71 \( 1 + 8.98iT - 71T^{2} \)
73 \( 1 + (11.2 + 11.2i)T + 73iT^{2} \)
79 \( 1 - 7.95T + 79T^{2} \)
83 \( 1 + (3.94 + 3.94i)T + 83iT^{2} \)
89 \( 1 + 9.30T + 89T^{2} \)
97 \( 1 + (-8.79 - 8.79i)T + 97iT^{2} \)
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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

−9.042508446577577690180010937102, −8.551228286432547627619089881622, −7.79562152417928784962909602275, −6.73196632677773853238447294975, −6.17820464600643413775682264653, −4.81879570440504837049405227272, −3.99458522993440805976199968295, −2.89147014243507900102373440525, −2.25832862076450187264729411007, −0.22966079033698631933562749785, 0.70042307782204370039495096567, 2.63836980969347015303447096215, 3.73511196575498511642436440301, 4.43426410659923874253191343166, 5.60660340778041485021782033535, 6.60659720682144090424151877810, 7.22048028610058824021520470459, 7.82488177716341990248076096815, 8.529597833365718924225049623421, 9.628588695830829692652907602959

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