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Properties

Label 2-187-17.4-c1-0-0
Degree $2$
Conductor $187$
Sign $-0.997 - 0.0655i$
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  + 0.357i·2-s + (−2.24 + 2.24i)3-s + 1.87·4-s + (−2.10 + 2.10i)5-s + (−0.802 − 0.802i)6-s + (−1.27 − 1.27i)7-s + 1.38i·8-s − 7.10i·9-s + (−0.751 − 0.751i)10-s + (0.707 + 0.707i)11-s + (−4.20 + 4.20i)12-s − 2.60·13-s + (0.455 − 0.455i)14-s − 9.46i·15-s + 3.25·16-s + (−4.11 + 0.234i)17-s + ⋯
L(s)  = 1  + 0.252i·2-s + (−1.29 + 1.29i)3-s + 0.936·4-s + (−0.941 + 0.941i)5-s + (−0.327 − 0.327i)6-s + (−0.481 − 0.481i)7-s + 0.488i·8-s − 2.36i·9-s + (−0.237 − 0.237i)10-s + (0.213 + 0.213i)11-s + (−1.21 + 1.21i)12-s − 0.723·13-s + (0.121 − 0.121i)14-s − 2.44i·15-s + 0.812·16-s + (−0.998 + 0.0568i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.997 - 0.0655i$
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: \((2,\ 187,\ (\ :1/2),\ -0.997 - 0.0655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0192695 + 0.586937i\)
\(L(\frac12)\) \(\approx\) \(0.0192695 + 0.586937i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.11 - 0.234i)T \)
good2 \( 1 - 0.357iT - 2T^{2} \)
3 \( 1 + (2.24 - 2.24i)T - 3iT^{2} \)
5 \( 1 + (2.10 - 2.10i)T - 5iT^{2} \)
7 \( 1 + (1.27 + 1.27i)T + 7iT^{2} \)
13 \( 1 + 2.60T + 13T^{2} \)
19 \( 1 - 5.05iT - 19T^{2} \)
23 \( 1 + (-2.99 - 2.99i)T + 23iT^{2} \)
29 \( 1 + (6.06 - 6.06i)T - 29iT^{2} \)
31 \( 1 + (-3.84 + 3.84i)T - 31iT^{2} \)
37 \( 1 + (-3.72 + 3.72i)T - 37iT^{2} \)
41 \( 1 + (-4.45 - 4.45i)T + 41iT^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 - 5.74T + 47T^{2} \)
53 \( 1 + 1.56iT - 53T^{2} \)
59 \( 1 - 3.86iT - 59T^{2} \)
61 \( 1 + (4.92 + 4.92i)T + 61iT^{2} \)
67 \( 1 + 6.00T + 67T^{2} \)
71 \( 1 + (1.40 - 1.40i)T - 71iT^{2} \)
73 \( 1 + (2.85 - 2.85i)T - 73iT^{2} \)
79 \( 1 + (-6.86 - 6.86i)T + 79iT^{2} \)
83 \( 1 + 6.73iT - 83T^{2} \)
89 \( 1 + 3.09T + 89T^{2} \)
97 \( 1 + (3.74 - 3.74i)T - 97iT^{2} \)
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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

−12.63277956753645315839239674959, −11.60074034497525797879421185940, −11.13806157774777726806044174774, −10.43222528165455559144871795009, −9.507431288420476305047906309069, −7.56019051453334657417269212023, −6.74211350424195395902966281513, −5.81481521409874667798159954799, −4.35391836409334622213767571957, −3.28782227688907774311920126974, 0.58656233405065583851931725921, 2.38446227678929659550712198512, 4.66940499344593024406923200074, 5.94100257151456471200906571325, 6.88666542060301385587268947539, 7.63484293043115947536622815494, 8.943766821528689972860229604891, 10.68180108337555230205795356138, 11.50959821995478262115738360397, 12.06736672581094502995573035832

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