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Properties

Label 2-187-17.16-c1-0-2
Degree $2$
Conductor $187$
Sign $0.981 - 0.192i$
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  − 2.61·2-s − 1.70i·3-s + 4.82·4-s + 0.846i·5-s + 4.45i·6-s + 3.51i·7-s − 7.37·8-s + 0.0885·9-s − 2.21i·10-s + i·11-s − 8.23i·12-s + 0.0408·13-s − 9.19i·14-s + 1.44·15-s + 9.62·16-s + (0.793 + 4.04i)17-s + ⋯
L(s)  = 1  − 1.84·2-s − 0.985i·3-s + 2.41·4-s + 0.378i·5-s + 1.81i·6-s + 1.33i·7-s − 2.60·8-s + 0.0295·9-s − 0.699i·10-s + 0.301i·11-s − 2.37i·12-s + 0.0113·13-s − 2.45i·14-s + 0.373·15-s + 2.40·16-s + (0.192 + 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.981 - 0.192i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.981 - 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.542120 + 0.0526609i\)
\(L(\frac12)\) \(\approx\) \(0.542120 + 0.0526609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - iT \)
17 \( 1 + (-0.793 - 4.04i)T \)
good2 \( 1 + 2.61T + 2T^{2} \)
3 \( 1 + 1.70iT - 3T^{2} \)
5 \( 1 - 0.846iT - 5T^{2} \)
7 \( 1 - 3.51iT - 7T^{2} \)
13 \( 1 - 0.0408T + 13T^{2} \)
19 \( 1 - 6.25T + 19T^{2} \)
23 \( 1 - 2.40iT - 23T^{2} \)
29 \( 1 + 9.14iT - 29T^{2} \)
31 \( 1 - 2.81iT - 31T^{2} \)
37 \( 1 - 1.51iT - 37T^{2} \)
41 \( 1 - 2.95iT - 41T^{2} \)
43 \( 1 + 8.15T + 43T^{2} \)
47 \( 1 + 0.595T + 47T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 + 6.48T + 59T^{2} \)
61 \( 1 - 6.01iT - 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 16.6iT - 71T^{2} \)
73 \( 1 + 6.52iT - 73T^{2} \)
79 \( 1 + 3.93iT - 79T^{2} \)
83 \( 1 - 5.74T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−12.05782412748592271145204324962, −11.72454225668220305284025521143, −10.36650759241920529582103574025, −9.529616793941690253077738216215, −8.509321887738304556904482325097, −7.70414245571219081732079420225, −6.81070225215119146739166021156, −5.82982515586056981164073922792, −2.72109713567822012604295537971, −1.51722658987686601780797286219, 1.01278856522464693889090598455, 3.35137442880925681546772360411, 5.00832760364910493883709486905, 6.90354008049675493991116274904, 7.59038149596026162430269809865, 8.867599587680352827276206820914, 9.567768686564644814365480726249, 10.40696420231762810781777226860, 10.92482079344570353093270017548, 12.05596479938982172422251879305

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