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Properties

Label 2-105-35.13-c3-0-19
Degree $2$
Conductor $105$
Sign $-0.191 + 0.981i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.36 − 3.36i)2-s + (−2.12 + 2.12i)3-s − 14.6i·4-s + (11.1 − 0.624i)5-s + 14.2i·6-s + (8.55 − 16.4i)7-s + (−22.5 − 22.5i)8-s − 8.99i·9-s + (35.5 − 39.7i)10-s − 19.8·11-s + (31.1 + 31.1i)12-s + (−10.0 + 10.0i)13-s + (−26.5 − 84.1i)14-s + (−22.3 + 25.0i)15-s − 34.4·16-s + (−42.0 − 42.0i)17-s + ⋯
L(s)  = 1  + (1.19 − 1.19i)2-s + (−0.408 + 0.408i)3-s − 1.83i·4-s + (0.998 − 0.0558i)5-s + 0.972i·6-s + (0.461 − 0.886i)7-s + (−0.997 − 0.997i)8-s − 0.333i·9-s + (1.12 − 1.25i)10-s − 0.543·11-s + (0.750 + 0.750i)12-s + (−0.214 + 0.214i)13-s + (−0.506 − 1.60i)14-s + (−0.384 + 0.430i)15-s − 0.537·16-s + (−0.599 − 0.599i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.191 + 0.981i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.191 + 0.981i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.84461 - 2.23868i\)
\(L(\frac12)\) \(\approx\) \(1.84461 - 2.23868i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.12 - 2.12i)T \)
5 \( 1 + (-11.1 + 0.624i)T \)
7 \( 1 + (-8.55 + 16.4i)T \)
good2 \( 1 + (-3.36 + 3.36i)T - 8iT^{2} \)
11 \( 1 + 19.8T + 1.33e3T^{2} \)
13 \( 1 + (10.0 - 10.0i)T - 2.19e3iT^{2} \)
17 \( 1 + (42.0 + 42.0i)T + 4.91e3iT^{2} \)
19 \( 1 - 0.840T + 6.85e3T^{2} \)
23 \( 1 + (-121. - 121. i)T + 1.21e4iT^{2} \)
29 \( 1 - 129. iT - 2.43e4T^{2} \)
31 \( 1 - 317. iT - 2.97e4T^{2} \)
37 \( 1 + (187. - 187. i)T - 5.06e4iT^{2} \)
41 \( 1 + 239. iT - 6.89e4T^{2} \)
43 \( 1 + (52.8 + 52.8i)T + 7.95e4iT^{2} \)
47 \( 1 + (-262. - 262. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-122. - 122. i)T + 1.48e5iT^{2} \)
59 \( 1 + 335.T + 2.05e5T^{2} \)
61 \( 1 + 883. iT - 2.26e5T^{2} \)
67 \( 1 + (537. - 537. i)T - 3.00e5iT^{2} \)
71 \( 1 + 691.T + 3.57e5T^{2} \)
73 \( 1 + (372. - 372. i)T - 3.89e5iT^{2} \)
79 \( 1 - 240. iT - 4.93e5T^{2} \)
83 \( 1 + (-490. + 490. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.62e3T + 7.04e5T^{2} \)
97 \( 1 + (-63.0 - 63.0i)T + 9.12e5iT^{2} \)
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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

−13.06855916075346592483131462962, −11.96054455333547553691745700776, −10.81352295327430469414386226970, −10.41466945059377279834025199974, −9.168398896471129510242939041559, −6.95647861717387797753807442020, −5.35897762584483295693021787889, −4.71421529266146323587964989221, −3.13538222064032485463276491791, −1.45110393573871074372853192798, 2.45506471858002337351876532289, 4.68868666944978423794147217002, 5.67033159518284723326135710409, 6.37066065982713721843206994295, 7.65146629335258431139834593900, 8.865927803099923678247306670651, 10.56223397111718459443806199280, 11.96605096472738058452272365216, 12.99832664035103129708198958039, 13.46360436052873786051224874735

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