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Properties

Label 2-1710-15.2-c1-0-2
Degree $2$
Conductor $1710$
Sign $-0.779 - 0.626i$
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 + (−0.379 − 2.20i)5-s + (2.44 + 2.44i)7-s + (0.707 + 0.707i)8-s + (1.82 + 1.29i)10-s + 1.85i·11-s + (−2.11 + 2.11i)13-s − 3.45·14-s − 1.00·16-s + (−2.29 + 2.29i)17-s + i·19-s + (−2.20 + 0.379i)20-s + (−1.31 − 1.31i)22-s + (−2.39 − 2.39i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.169 − 0.985i)5-s + (0.924 + 0.924i)7-s + (0.250 + 0.250i)8-s + (0.577 + 0.408i)10-s + 0.559i·11-s + (−0.587 + 0.587i)13-s − 0.924·14-s − 0.250·16-s + (−0.555 + 0.555i)17-s + 0.229i·19-s + (−0.492 + 0.0847i)20-s + (−0.279 − 0.279i)22-s + (−0.498 − 0.498i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.779 - 0.626i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.779 - 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7327270982\)
\(L(\frac12)\) \(\approx\) \(0.7327270982\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.379 + 2.20i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \)
11 \( 1 - 1.85iT - 11T^{2} \)
13 \( 1 + (2.11 - 2.11i)T - 13iT^{2} \)
17 \( 1 + (2.29 - 2.29i)T - 17iT^{2} \)
23 \( 1 + (2.39 + 2.39i)T + 23iT^{2} \)
29 \( 1 + 6.09T + 29T^{2} \)
31 \( 1 - 1.81T + 31T^{2} \)
37 \( 1 + (-4.38 - 4.38i)T + 37iT^{2} \)
41 \( 1 + 4.05iT - 41T^{2} \)
43 \( 1 + (7.27 - 7.27i)T - 43iT^{2} \)
47 \( 1 + (-1.44 + 1.44i)T - 47iT^{2} \)
53 \( 1 + (-1.03 - 1.03i)T + 53iT^{2} \)
59 \( 1 + 3.94T + 59T^{2} \)
61 \( 1 + 2.18T + 61T^{2} \)
67 \( 1 + (-4.80 - 4.80i)T + 67iT^{2} \)
71 \( 1 + 2.06iT - 71T^{2} \)
73 \( 1 + (9.58 - 9.58i)T - 73iT^{2} \)
79 \( 1 - 1.43iT - 79T^{2} \)
83 \( 1 + (-2.48 - 2.48i)T + 83iT^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + (-4 - 4i)T + 97iT^{2} \)
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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

−9.431825179114574640100813710278, −8.723451776433843766304108790783, −8.198625857121635586179565068051, −7.49240759969008997824434016705, −6.44395757695169634102393118159, −5.55365456647278359671473586029, −4.82529968374264569447300665614, −4.15461767113206489700782234166, −2.29097742207481672160203707270, −1.50941271433777624855962273914, 0.32087444081319070216045299292, 1.82277010384345366730217934844, 2.89013173529579584579441069913, 3.78616991162789756890573143652, 4.69377942909195354264507008148, 5.83030488703450273860198131615, 6.96035994167335214511955003572, 7.55583319014158132864284773726, 8.078323059863278814734203682475, 9.113750479087470612813566415540

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