Properties

Label 2-190-5.4-c1-0-5
Degree $2$
Conductor $190$
Sign $0.783 + 0.621i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + 1.36i·3-s − 4-s + (1.38 − 1.75i)5-s + 1.36·6-s − 0.636i·7-s + i·8-s + 1.14·9-s + (−1.75 − 1.38i)10-s + 3.50·11-s − 1.36i·12-s − 0.141i·13-s − 0.636·14-s + (2.38 + 1.89i)15-s + 16-s − 2.14i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.787i·3-s − 0.5·4-s + (0.621 − 0.783i)5-s + 0.556·6-s − 0.240i·7-s + 0.353i·8-s + 0.380·9-s + (−0.554 − 0.439i)10-s + 1.05·11-s − 0.393i·12-s − 0.0391i·13-s − 0.170·14-s + (0.616 + 0.488i)15-s + 0.250·16-s − 0.519i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ 0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21683 - 0.423776i\)
\(L(\frac12)\) \(\approx\) \(1.21683 - 0.423776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-1.38 + 1.75i)T \)
19 \( 1 + T \)
good3 \( 1 - 1.36iT - 3T^{2} \)
7 \( 1 + 0.636iT - 7T^{2} \)
11 \( 1 - 3.50T + 11T^{2} \)
13 \( 1 + 0.141iT - 13T^{2} \)
17 \( 1 + 2.14iT - 17T^{2} \)
23 \( 1 - 4.91iT - 23T^{2} \)
29 \( 1 + 7.15T + 29T^{2} \)
31 \( 1 + 7.78T + 31T^{2} \)
37 \( 1 + 3.27iT - 37T^{2} \)
41 \( 1 + 4.23T + 41T^{2} \)
43 \( 1 - 2.49iT - 43T^{2} \)
47 \( 1 - 10.2iT - 47T^{2} \)
53 \( 1 - 8.14iT - 53T^{2} \)
59 \( 1 - 5.64T + 59T^{2} \)
61 \( 1 + 6.49T + 61T^{2} \)
67 \( 1 + 8.37iT - 67T^{2} \)
71 \( 1 + 8.95T + 71T^{2} \)
73 \( 1 + 3.69iT - 73T^{2} \)
79 \( 1 - 4.17T + 79T^{2} \)
83 \( 1 + 9.00iT - 83T^{2} \)
89 \( 1 - 6.77T + 89T^{2} \)
97 \( 1 + 14.5iT - 97T^{2} \)
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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

−12.45355914397413285174338145396, −11.38056419530427717937753207683, −10.42791842075972560550982798719, −9.289398527533094864208131855368, −9.198291178610722541948875229885, −7.43090163882148234392101774312, −5.77390626129074940428910387873, −4.62643853548956378579857939981, −3.66373112377009733709320178132, −1.59486006614633122519785379543, 1.90255614503579768851563293398, 3.86729568264320067947768012528, 5.62401297475751619080272198727, 6.63990066743912451584119523402, 7.17215960297776972529262274790, 8.503655285181399809546950927886, 9.555164455432317290604680722520, 10.60255783512724041641928729540, 11.87078147064312528679265012519, 12.90356667263701116500892765978

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