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Properties

Label 2-187-11.9-c1-0-7
Degree $2$
Conductor $187$
Sign $0.944 + 0.329i$
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  + (0.400 − 1.23i)2-s + (−0.386 + 0.280i)3-s + (0.261 + 0.189i)4-s + (0.547 + 1.68i)5-s + (0.190 + 0.587i)6-s + (1.85 + 1.34i)7-s + (2.43 − 1.76i)8-s + (−0.856 + 2.63i)9-s + 2.29·10-s + (−2.54 − 2.12i)11-s − 0.154·12-s + (1.04 − 3.20i)13-s + (2.40 − 1.74i)14-s + (−0.684 − 0.497i)15-s + (−1.00 − 3.09i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.282 − 0.870i)2-s + (−0.222 + 0.161i)3-s + (0.130 + 0.0949i)4-s + (0.244 + 0.753i)5-s + (0.0779 + 0.239i)6-s + (0.701 + 0.509i)7-s + (0.860 − 0.625i)8-s + (−0.285 + 0.878i)9-s + 0.725·10-s + (−0.767 − 0.641i)11-s − 0.0445·12-s + (0.289 − 0.889i)13-s + (0.642 − 0.466i)14-s + (−0.176 − 0.128i)15-s + (−0.251 − 0.772i)16-s + (−0.0749 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.944 + 0.329i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45661 - 0.247149i\)
\(L(\frac12)\) \(\approx\) \(1.45661 - 0.247149i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.54 + 2.12i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (-0.400 + 1.23i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.386 - 0.280i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.547 - 1.68i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.85 - 1.34i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.04 + 3.20i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (-0.372 + 0.270i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.90T + 23T^{2} \)
29 \( 1 + (1.39 + 1.01i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.12 - 3.47i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.14 - 0.835i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.34 - 1.70i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.72T + 43T^{2} \)
47 \( 1 + (-7.98 + 5.80i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.36 - 13.4i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (11.4 + 8.33i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.27 - 3.92i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 3.30T + 67T^{2} \)
71 \( 1 + (2.96 + 9.11i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.46 - 5.42i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.84 - 5.67i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.10 + 12.6i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 5.59T + 89T^{2} \)
97 \( 1 + (1.35 - 4.16i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.34288699939443586494388815388, −11.38090821261730667635391848659, −10.73792883077437665005777997420, −10.17019095476463517676120936800, −8.378573232059888664028491958320, −7.55361122469646995288071323749, −5.99667222373700420056163202343, −4.88482437381033306722755597890, −3.19771637676085693303937277010, −2.19263888269656206054499470956, 1.71306109526390893919468878460, 4.27066975831041600302913018118, 5.30051348619421521043856270681, 6.31550281963709584500147484524, 7.36862724852139777955553643674, 8.331644469571745901463648173741, 9.552011757494261253348402869032, 10.78110828948694724834591137883, 11.71439782100049923100188312045, 12.76353678877368047704244347410

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