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Properties

Label 2-1710-15.8-c1-0-9
Degree $2$
Conductor $1710$
Sign $-0.144 - 0.989i$
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 + (−1.70 − 1.44i)5-s + (0.414 − 0.414i)7-s + (−0.707 + 0.707i)8-s + (−0.185 − 2.22i)10-s + 2.58i·11-s + (−2.88 − 2.88i)13-s + 0.585·14-s − 1.00·16-s + (5.60 + 5.60i)17-s + i·19-s + (1.44 − 1.70i)20-s + (−1.82 + 1.82i)22-s + (0.322 − 0.322i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.763 − 0.645i)5-s + (0.156 − 0.156i)7-s + (−0.250 + 0.250i)8-s + (−0.0587 − 0.704i)10-s + 0.779i·11-s + (−0.801 − 0.801i)13-s + 0.156·14-s − 0.250·16-s + (1.36 + 1.36i)17-s + 0.229i·19-s + (0.322 − 0.381i)20-s + (−0.389 + 0.389i)22-s + (0.0673 − 0.0673i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.634728658\)
\(L(\frac12)\) \(\approx\) \(1.634728658\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 1.44i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-0.414 + 0.414i)T - 7iT^{2} \)
11 \( 1 - 2.58iT - 11T^{2} \)
13 \( 1 + (2.88 + 2.88i)T + 13iT^{2} \)
17 \( 1 + (-5.60 - 5.60i)T + 17iT^{2} \)
23 \( 1 + (-0.322 + 0.322i)T - 23iT^{2} \)
29 \( 1 - 5.77T + 29T^{2} \)
31 \( 1 + 5.13T + 31T^{2} \)
37 \( 1 + (1.41 - 1.41i)T - 37iT^{2} \)
41 \( 1 - 5.47iT - 41T^{2} \)
43 \( 1 + (-2.47 - 2.47i)T + 43iT^{2} \)
47 \( 1 + (-9.45 - 9.45i)T + 47iT^{2} \)
53 \( 1 + (4.24 - 4.24i)T - 53iT^{2} \)
59 \( 1 - 0.302T + 59T^{2} \)
61 \( 1 - 1.25T + 61T^{2} \)
67 \( 1 + (-4.30 + 4.30i)T - 67iT^{2} \)
71 \( 1 - 7.25iT - 71T^{2} \)
73 \( 1 + (1.82 + 1.82i)T + 73iT^{2} \)
79 \( 1 - 2.61iT - 79T^{2} \)
83 \( 1 + (-5.45 + 5.45i)T - 83iT^{2} \)
89 \( 1 + 7.43T + 89T^{2} \)
97 \( 1 + (6.23 - 6.23i)T - 97iT^{2} \)
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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

−9.483772434400052406783223357095, −8.475875118925986862840490292652, −7.74650283204344074612155578966, −7.46160796852631665124684089226, −6.23274722833820964168609154032, −5.38564956094928921140459275261, −4.62716232800631871189312389932, −3.88546103137038238990265079400, −2.87270475536537822015484124224, −1.27451845574076676587608614906, 0.57382246165070478779006531283, 2.26587287078861357525870995790, 3.14724805119734103316840303134, 3.90975462724820163939800482527, 4.95989020606652820475778878729, 5.65760434549551029484279875490, 6.88247046386244857184599563256, 7.31228626366400702024643602929, 8.349088284322329198928725467763, 9.219505669826842761131119172751

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