Properties

Label 2-187-17.4-c1-0-12
Degree $2$
Conductor $187$
Sign $-0.615 + 0.788i$
Analytic cond. $1.49320$
Root an. cond. $1.22196$
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  − 2i·2-s + (0.414 − 0.414i)3-s − 2·4-s + (2 − 2i)5-s + (−0.828 − 0.828i)6-s + (0.707 + 0.707i)7-s + 2.65i·9-s + (−4 − 4i)10-s + (0.707 + 0.707i)11-s + (−0.828 + 0.828i)12-s − 3.41·13-s + (1.41 − 1.41i)14-s − 1.65i·15-s − 4·16-s + (−2.12 + 3.53i)17-s + 5.31·18-s + ⋯
L(s)  = 1  − 1.41i·2-s + (0.239 − 0.239i)3-s − 4-s + (0.894 − 0.894i)5-s + (−0.338 − 0.338i)6-s + (0.267 + 0.267i)7-s + 0.885i·9-s + (−1.26 − 1.26i)10-s + (0.213 + 0.213i)11-s + (−0.239 + 0.239i)12-s − 0.946·13-s + (0.377 − 0.377i)14-s − 0.427i·15-s − 16-s + (−0.514 + 0.857i)17-s + 1.25·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ -0.615 + 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613933 - 1.25824i\)
\(L(\frac12)\) \(\approx\) \(0.613933 - 1.25824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (2.12 - 3.53i)T \)
good2 \( 1 + 2iT - 2T^{2} \)
3 \( 1 + (-0.414 + 0.414i)T - 3iT^{2} \)
5 \( 1 + (-2 + 2i)T - 5iT^{2} \)
7 \( 1 + (-0.707 - 0.707i)T + 7iT^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
19 \( 1 + 2.58iT - 19T^{2} \)
23 \( 1 + (-3 - 3i)T + 23iT^{2} \)
29 \( 1 + (-2.70 + 2.70i)T - 29iT^{2} \)
31 \( 1 + (4.82 - 4.82i)T - 31iT^{2} \)
37 \( 1 + (-6.65 + 6.65i)T - 37iT^{2} \)
41 \( 1 + (3.53 + 3.53i)T + 41iT^{2} \)
43 \( 1 - 5.41iT - 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + 3.48iT - 53T^{2} \)
59 \( 1 - 4.65iT - 59T^{2} \)
61 \( 1 + (4.82 + 4.82i)T + 61iT^{2} \)
67 \( 1 - T + 67T^{2} \)
71 \( 1 + (11.0 - 11.0i)T - 71iT^{2} \)
73 \( 1 + (5.77 - 5.77i)T - 73iT^{2} \)
79 \( 1 + (2.24 + 2.24i)T + 79iT^{2} \)
83 \( 1 + 4.82iT - 83T^{2} \)
89 \( 1 + 1.82T + 89T^{2} \)
97 \( 1 + (5.41 - 5.41i)T - 97iT^{2} \)
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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

−12.35149419629317010109401111385, −11.26157782541037910922244320253, −10.35846158080337755829573561442, −9.395019140298678483355718185016, −8.668468471240592100767155510920, −7.18254722280278051557913773539, −5.44940931578086424644632342306, −4.41684876486692599532384874125, −2.53999336474189481968251595350, −1.60794690122680362463898630564, 2.71390505954803253877501966587, 4.56947131096828823383902228027, 5.89886161724425988781611885570, 6.70639466026874918149851272007, 7.51352702148216581048976856627, 8.865900400970871374476584102713, 9.673022120572290177294443749598, 10.76968688049761281685063967111, 11.99512380713539577350558112792, 13.46793635401235386985670606433

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