↑

Properties

Label 2-105-35.17-c1-0-7
Degree $2$
Conductor $105$
Sign $-0.900 + 0.435i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
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  + (0.650 − 2.42i)2-s + (−0.965 + 0.258i)3-s + (−3.73 − 2.15i)4-s + (−1.42 − 1.72i)5-s + 2.51i·6-s + (2.09 + 1.61i)7-s + (−4.11 + 4.11i)8-s + (0.866 − 0.499i)9-s + (−5.11 + 2.33i)10-s + (2.73 − 4.74i)11-s + (4.17 + 1.11i)12-s + (0.579 + 0.579i)13-s + (5.29 − 4.02i)14-s + (1.82 + 1.29i)15-s + (3.00 + 5.19i)16-s + (1.22 + 4.58i)17-s + ⋯
L(s)  = 1  + (0.459 − 1.71i)2-s + (−0.557 + 0.149i)3-s + (−1.86 − 1.07i)4-s + (−0.636 − 0.770i)5-s + 1.02i·6-s + (0.791 + 0.611i)7-s + (−1.45 + 1.45i)8-s + (0.288 − 0.166i)9-s + (−1.61 + 0.738i)10-s + (0.825 − 1.42i)11-s + (1.20 + 0.322i)12-s + (0.160 + 0.160i)13-s + (1.41 − 1.07i)14-s + (0.470 + 0.334i)15-s + (0.750 + 1.29i)16-s + (0.298 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.900 + 0.435i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218094 - 0.950880i\)
\(L(\frac12)\) \(\approx\) \(0.218094 - 0.950880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (1.42 + 1.72i)T \)
7 \( 1 + (-2.09 - 1.61i)T \)
good2 \( 1 + (-0.650 + 2.42i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-2.73 + 4.74i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.579 - 0.579i)T + 13iT^{2} \)
17 \( 1 + (-1.22 - 4.58i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.220 + 0.381i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.70 - 0.457i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.853iT - 29T^{2} \)
31 \( 1 + (-2.32 - 1.34i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0668 + 0.249i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.321iT - 41T^{2} \)
43 \( 1 + (0.631 - 0.631i)T - 43iT^{2} \)
47 \( 1 + (7.91 + 2.12i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.96 - 11.0i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.89 + 5.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.73 - 3.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.16 + 1.38i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 + (-8.53 + 2.28i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.02 + 5.20i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.47 + 8.47i)T + 83iT^{2} \)
89 \( 1 + (4.03 + 6.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.99 - 5.99i)T - 97iT^{2} \)
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

−12.89383897680735333644962614883, −11.99554265806325398996047518876, −11.44296034889184791220470558197, −10.67883170662961381322030353124, −9.172228488417950282857229272634, −8.372318754316292584352186248096, −5.82550276455525929554203244372, −4.66819238482676894346167200196, −3.54070085081698140491733248153, −1.29879880467418496892090275624, 4.09686038874895002262265543361, 5.04888257586902691612291363602, 6.65950167067901538133188612668, 7.21590939458234771676450666896, 8.119616569826658405536904611353, 9.786099610965789169930352113416, 11.27951780373636329552966341759, 12.28693204477732445029938453028, 13.61116471595278597212446818884, 14.55079569970153447196401938221

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