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Properties

Label 2-187-11.3-c1-0-10
Degree $2$
Conductor $187$
Sign $0.991 - 0.128i$
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  + (1.61 − 1.17i)2-s + (1.01 + 3.12i)3-s + (0.617 − 1.90i)4-s + (−1.06 − 0.773i)5-s + (5.30 + 3.85i)6-s + (1.03 − 3.19i)7-s + (0.00105 + 0.00324i)8-s + (−6.28 + 4.56i)9-s − 2.63·10-s + (−3.27 + 0.505i)11-s + 6.55·12-s + (2.66 − 1.93i)13-s + (−2.07 − 6.38i)14-s + (1.33 − 4.10i)15-s + (3.23 + 2.35i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (1.14 − 0.831i)2-s + (0.585 + 1.80i)3-s + (0.308 − 0.950i)4-s + (−0.476 − 0.346i)5-s + (2.16 + 1.57i)6-s + (0.392 − 1.20i)7-s + (0.000372 + 0.00114i)8-s + (−2.09 + 1.52i)9-s − 0.832·10-s + (−0.988 + 0.152i)11-s + 1.89·12-s + (0.740 − 0.537i)13-s + (−0.554 − 1.70i)14-s + (0.344 − 1.06i)15-s + (0.809 + 0.588i)16-s + (−0.196 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.991 - 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16979 + 0.140523i\)
\(L(\frac12)\) \(\approx\) \(2.16979 + 0.140523i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3.27 - 0.505i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-1.61 + 1.17i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.01 - 3.12i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.06 + 0.773i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.03 + 3.19i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.66 + 1.93i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (1.13 + 3.49i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.473T + 23T^{2} \)
29 \( 1 + (0.409 - 1.26i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.89 - 5.01i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.688 + 2.12i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.88 - 8.89i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.41T + 43T^{2} \)
47 \( 1 + (-0.270 - 0.833i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.82 + 3.50i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.73 - 11.4i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.79 + 2.75i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.07T + 67T^{2} \)
71 \( 1 + (-11.0 - 8.03i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.198 + 0.609i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.21 - 5.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.48 + 1.80i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 4.49T + 89T^{2} \)
97 \( 1 + (8.00 - 5.81i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.79693851275976793856954371786, −11.23416551771722314064649061263, −10.83805838470470346310552407981, −10.12118393063607238129233718914, −8.722523699996637561110345109719, −7.80142191388997005119449677721, −5.40125145665752824873436498639, −4.54395610090769079127626445338, −3.91148824284512165965469850746, −2.83810671256986141764687473925, 2.16127060035873374347538557584, 3.56532219427049624014606474805, 5.55220885093932393457466150528, 6.20946713302951481488040642668, 7.35263509550985368570595713987, 7.989174471392546344930538935538, 8.984240114585896150178066424658, 11.18620147589064008999468270887, 12.14246510041958844669876986040, 12.80702237272860583990938702893

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