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Properties

Label 2-1710-95.37-c1-0-48
Degree $2$
Conductor $1710$
Sign $-0.995 - 0.0909i$
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 + (1.66 − 1.49i)5-s + (−0.170 − 0.170i)7-s + (−0.707 − 0.707i)8-s + (0.120 − 2.23i)10-s − 3.43·11-s + (−4.54 − 4.54i)13-s − 0.240·14-s − 1.00·16-s + (−0.537 − 0.537i)17-s + (−2.42 + 3.62i)19-s + (−1.49 − 1.66i)20-s + (−2.42 + 2.42i)22-s + (−5.15 + 5.15i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.744 − 0.668i)5-s + (−0.0643 − 0.0643i)7-s + (−0.250 − 0.250i)8-s + (0.0380 − 0.706i)10-s − 1.03·11-s + (−1.25 − 1.25i)13-s − 0.0643·14-s − 0.250·16-s + (−0.130 − 0.130i)17-s + (−0.556 + 0.830i)19-s + (−0.334 − 0.372i)20-s + (−0.517 + 0.517i)22-s + (−1.07 + 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.995 - 0.0909i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.995 - 0.0909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.217229008\)
\(L(\frac12)\) \(\approx\) \(1.217229008\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 + 1.49i)T \)
19 \( 1 + (2.42 - 3.62i)T \)
good7 \( 1 + (0.170 + 0.170i)T + 7iT^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
13 \( 1 + (4.54 + 4.54i)T + 13iT^{2} \)
17 \( 1 + (0.537 + 0.537i)T + 17iT^{2} \)
23 \( 1 + (5.15 - 5.15i)T - 23iT^{2} \)
29 \( 1 + 5.37T + 29T^{2} \)
31 \( 1 + 5.37iT - 31T^{2} \)
37 \( 1 + (-5.54 + 5.54i)T - 37iT^{2} \)
41 \( 1 - 3.68iT - 41T^{2} \)
43 \( 1 + (-2.29 + 2.29i)T - 43iT^{2} \)
47 \( 1 + (-9.57 - 9.57i)T + 47iT^{2} \)
53 \( 1 + (1.93 + 1.93i)T + 53iT^{2} \)
59 \( 1 + 4.20T + 59T^{2} \)
61 \( 1 - 1.65T + 61T^{2} \)
67 \( 1 + (0.481 - 0.481i)T - 67iT^{2} \)
71 \( 1 + 8.93iT - 71T^{2} \)
73 \( 1 + (-8.74 + 8.74i)T - 73iT^{2} \)
79 \( 1 + 1.15T + 79T^{2} \)
83 \( 1 + (-9.97 + 9.97i)T - 83iT^{2} \)
89 \( 1 - 2.10T + 89T^{2} \)
97 \( 1 + (10.8 - 10.8i)T - 97iT^{2} \)
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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

−9.245618027624524735191064188852, −7.936331506321392661012994068686, −7.60706067322258099112690911907, −6.04846303934203244344818103176, −5.60668994754897063465167970982, −4.87682683959997899495124716078, −3.88619403909598355068382867926, −2.66058134022345522781771582210, −1.93832251319381497314924193079, −0.34093878768750934948300644557, 2.20140378502719700896115786666, 2.66440916951185916480504809615, 4.08097558811454046810463318067, 4.90345571066629513494100624914, 5.71967804379917044027648212661, 6.62391873969077677511661476873, 7.09469489484122488584120353200, 7.986813578724313609302541871584, 8.973679826538100476666745619984, 9.693098086776264058699578078415

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