↑

Properties

Label 2-1710-95.37-c1-0-30
Degree $2$
Conductor $1710$
Sign $0.308 + 0.951i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.114 + 2.23i)5-s + (−1.40 − 1.40i)7-s + (−0.707 − 0.707i)8-s + (1.49 + 1.66i)10-s + 5.29·11-s + (−1.91 − 1.91i)13-s − 1.98·14-s − 1.00·16-s + (−0.488 − 0.488i)17-s + (−2.44 − 3.60i)19-s + (2.23 + 0.114i)20-s + (3.74 − 3.74i)22-s + (3.88 − 3.88i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.0512 + 0.998i)5-s + (−0.530 − 0.530i)7-s + (−0.250 − 0.250i)8-s + (0.473 + 0.524i)10-s + 1.59·11-s + (−0.532 − 0.532i)13-s − 0.530·14-s − 0.250·16-s + (−0.118 − 0.118i)17-s + (−0.561 − 0.827i)19-s + (0.499 + 0.0256i)20-s + (0.797 − 0.797i)22-s + (0.810 − 0.810i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.308 + 0.951i$
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: \((2,\ 1710,\ (\ :1/2),\ 0.308 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.116309341\)
\(L(\frac12)\) \(\approx\) \(2.116309341\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.114 - 2.23i)T \)
19 \( 1 + (2.44 + 3.60i)T \)
good7 \( 1 + (1.40 + 1.40i)T + 7iT^{2} \)
11 \( 1 - 5.29T + 11T^{2} \)
13 \( 1 + (1.91 + 1.91i)T + 13iT^{2} \)
17 \( 1 + (0.488 + 0.488i)T + 17iT^{2} \)
23 \( 1 + (-3.88 + 3.88i)T - 23iT^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 - 7.84iT - 31T^{2} \)
37 \( 1 + (-5.60 + 5.60i)T - 37iT^{2} \)
41 \( 1 - 1.90iT - 41T^{2} \)
43 \( 1 + (-7.12 + 7.12i)T - 43iT^{2} \)
47 \( 1 + (-7.86 - 7.86i)T + 47iT^{2} \)
53 \( 1 + (8.93 + 8.93i)T + 53iT^{2} \)
59 \( 1 - 0.611T + 59T^{2} \)
61 \( 1 - 8.31T + 61T^{2} \)
67 \( 1 + (-10.2 + 10.2i)T - 67iT^{2} \)
71 \( 1 + 3.97iT - 71T^{2} \)
73 \( 1 + (-7.72 + 7.72i)T - 73iT^{2} \)
79 \( 1 + 17.1T + 79T^{2} \)
83 \( 1 + (5.43 - 5.43i)T - 83iT^{2} \)
89 \( 1 + 5.42T + 89T^{2} \)
97 \( 1 + (3.10 - 3.10i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.393357524859176396181314565592, −8.534968508209237495771119030240, −7.19434700181735280265159281379, −6.75561160006386240711133793813, −6.11989156044870850419464059069, −4.83858245739599350237926404314, −4.00177678581888097414905329937, −3.18533563746248676482612595000, −2.34852994440374563240053622497, −0.77116111885728919167118354560, 1.25654858657748579906964201179, 2.62490947192484929735578389345, 3.99911818553920615604946833593, 4.37613825447472765488159191779, 5.54830769690448438425882872487, 6.18313300775280132740758124738, 6.93188582841540082257608565501, 7.931752014367061239742636487419, 8.744012960160732110989062756335, 9.330126873427479129892271007792

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