↑

Properties

Label 2-190-95.48-c1-0-3
Degree $2$
Conductor $190$
Sign $0.375 - 0.926i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
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  + (−0.573 + 0.819i)2-s + (−0.0709 + 0.810i)3-s + (−0.342 − 0.939i)4-s + (2.23 + 0.0422i)5-s + (−0.623 − 0.523i)6-s + (−0.211 − 0.790i)7-s + (0.965 + 0.258i)8-s + (2.30 + 0.405i)9-s + (−1.31 + 1.80i)10-s + (2.32 + 4.02i)11-s + (0.785 − 0.210i)12-s + (−2.88 + 0.251i)13-s + (0.769 + 0.279i)14-s + (−0.192 + 1.80i)15-s + (−0.766 + 0.642i)16-s + (−5.85 − 4.09i)17-s + ⋯
L(s)  = 1  + (−0.405 + 0.579i)2-s + (−0.0409 + 0.467i)3-s + (−0.171 − 0.469i)4-s + (0.999 + 0.0188i)5-s + (−0.254 − 0.213i)6-s + (−0.0800 − 0.298i)7-s + (0.341 + 0.0915i)8-s + (0.767 + 0.135i)9-s + (−0.416 + 0.571i)10-s + (0.701 + 1.21i)11-s + (0.226 − 0.0607i)12-s + (−0.798 + 0.0698i)13-s + (0.205 + 0.0748i)14-s + (−0.0497 + 0.467i)15-s + (−0.191 + 0.160i)16-s + (−1.41 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.375 - 0.926i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ 0.375 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.925731 + 0.623785i\)
\(L(\frac12)\) \(\approx\) \(0.925731 + 0.623785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.573 - 0.819i)T \)
5 \( 1 + (-2.23 - 0.0422i)T \)
19 \( 1 + (-1.08 - 4.22i)T \)
good3 \( 1 + (0.0709 - 0.810i)T + (-2.95 - 0.520i)T^{2} \)
7 \( 1 + (0.211 + 0.790i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.32 - 4.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.88 - 0.251i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (5.85 + 4.09i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (-5.90 - 2.75i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-1.50 + 8.53i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (6.69 + 3.86i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 + (3.03 + 3.61i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.67 + 3.59i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (5.50 + 7.85i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (-2.45 + 5.27i)T + (-34.0 - 40.6i)T^{2} \)
59 \( 1 + (0.124 + 0.708i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (8.02 - 2.92i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (8.69 - 6.08i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-1.92 + 5.27i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-1.99 - 0.174i)T + (71.8 + 12.6i)T^{2} \)
79 \( 1 + (8.08 - 6.78i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.46 - 1.19i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.66 + 1.39i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (2.55 - 3.65i)T + (-33.1 - 91.1i)T^{2} \)
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

−12.99685318518027292770639874960, −11.65508267468767858471082857891, −10.30451436672940139338668463987, −9.704654394512517191471971878866, −9.094746734070410037702740360108, −7.34088720936631152963180283145, −6.76705513737338624493468304110, −5.27384415782438121713519983027, −4.30525942678192729206802895705, −1.96112111832224730434365888822, 1.45238675012349071956553066463, 2.92462387442726681717493341822, 4.73244329093576717351507255833, 6.27679946516311396892625782669, 7.11446663905624470443994855143, 8.806229532391021287357315543344, 9.197488126419474992348181911843, 10.52819311281707261235570281582, 11.21513044696459096567900217365, 12.62653853601224180266191762613

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