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Properties

Label 2-187-11.5-c1-0-4
Degree $2$
Conductor $187$
Sign $0.443 - 0.896i$
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  + (0.282 + 0.868i)2-s + (1.40 + 1.01i)3-s + (0.943 − 0.685i)4-s + (−0.0971 + 0.298i)5-s + (−0.488 + 1.50i)6-s + (−2.74 + 1.99i)7-s + (2.33 + 1.69i)8-s + (−0.00124 − 0.00383i)9-s − 0.287·10-s + (0.331 − 3.30i)11-s + 2.01·12-s + (0.0768 + 0.236i)13-s + (−2.50 − 1.81i)14-s + (−0.440 + 0.319i)15-s + (−0.0950 + 0.292i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.199 + 0.614i)2-s + (0.808 + 0.587i)3-s + (0.471 − 0.342i)4-s + (−0.0434 + 0.133i)5-s + (−0.199 + 0.613i)6-s + (−1.03 + 0.752i)7-s + (0.826 + 0.600i)8-s + (−0.000415 − 0.00127i)9-s − 0.0907·10-s + (0.0999 − 0.994i)11-s + 0.582·12-s + (0.0213 + 0.0656i)13-s + (−0.668 − 0.486i)14-s + (−0.113 + 0.0825i)15-s + (−0.0237 + 0.0731i)16-s + (0.0749 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.443 - 0.896i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.443 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43561 + 0.890883i\)
\(L(\frac12)\) \(\approx\) \(1.43561 + 0.890883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.331 + 3.30i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (-0.282 - 0.868i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.40 - 1.01i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.0971 - 0.298i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.74 - 1.99i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.0768 - 0.236i)T + (-10.5 + 7.64i)T^{2} \)
19 \( 1 + (0.313 + 0.227i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6.93T + 23T^{2} \)
29 \( 1 + (7.30 - 5.30i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.50 + 4.62i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-8.38 + 6.09i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.60 + 2.61i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.40T + 43T^{2} \)
47 \( 1 + (5.80 + 4.21i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.43 - 10.5i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.38 - 2.45i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.447 + 1.37i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 0.581T + 67T^{2} \)
71 \( 1 + (-0.234 + 0.721i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.23 - 5.98i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.28 - 10.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.32 + 7.16i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 3.66T + 89T^{2} \)
97 \( 1 + (-5.42 - 16.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.97622248169725829805730520041, −11.72727041059148969176787438190, −10.69951230117734815318176942694, −9.556429480851637813995110789056, −8.855216660652481929725342680098, −7.61852742759435393942838548037, −6.32311730587023660683157734002, −5.60874979941966744849645356607, −3.78704481226530147677610989708, −2.60393859270255800971022787773, 1.90485824084399678120753786701, 3.15134199820151480551073167479, 4.29347223887650737987091253618, 6.44180128741645885834690507725, 7.36525919874168668738287519047, 8.138353241105816940214174053873, 9.657077308270667492479682383272, 10.39307343228400262727531018695, 11.61688359469559586154475898956, 12.72642902405760244881732349425

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