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Properties

Label 2-187-11.5-c1-0-10
Degree $2$
Conductor $187$
Sign $-0.649 + 0.760i$
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  + (−0.5 − 1.53i)2-s + (−0.0922 − 0.0670i)3-s + (−0.5 + 0.363i)4-s + (−0.0352 + 0.108i)5-s + (−0.0570 + 0.175i)6-s + (1.09 − 0.793i)7-s + (−1.80 − 1.31i)8-s + (−0.923 − 2.84i)9-s + 0.184·10-s + (1.12 − 3.11i)11-s + 0.0704·12-s + (0.690 + 2.12i)13-s + (−1.76 − 1.28i)14-s + (0.0105 − 0.00764i)15-s + (−1.50 + 4.61i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.353 − 1.08i)2-s + (−0.0532 − 0.0386i)3-s + (−0.250 + 0.181i)4-s + (−0.0157 + 0.0484i)5-s + (−0.0232 + 0.0716i)6-s + (0.412 − 0.299i)7-s + (−0.639 − 0.464i)8-s + (−0.307 − 0.946i)9-s + 0.0583·10-s + (0.339 − 0.940i)11-s + 0.0203·12-s + (0.191 + 0.589i)13-s + (−0.472 − 0.343i)14-s + (0.00271 − 0.00197i)15-s + (−0.375 + 1.15i)16-s + (−0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.649 + 0.760i$
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: \((2,\ 187,\ (\ :1/2),\ -0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.411272 - 0.892025i\)
\(L(\frac12)\) \(\approx\) \(0.411272 - 0.892025i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-1.12 + 3.11i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (0.5 + 1.53i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.0922 + 0.0670i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.0352 - 0.108i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.09 + 0.793i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.690 - 2.12i)T + (-10.5 + 7.64i)T^{2} \)
19 \( 1 + (3.30 + 2.39i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.18T + 23T^{2} \)
29 \( 1 + (-0.582 + 0.423i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.35 - 7.23i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.56 + 4.76i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.20 + 5.23i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.74T + 43T^{2} \)
47 \( 1 + (-7.96 - 5.78i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.0435 - 0.134i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.42 + 1.76i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.36 - 13.4i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.45T + 67T^{2} \)
71 \( 1 + (-1.26 + 3.89i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.51 - 4.00i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.67 + 14.3i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.925 - 2.84i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (0.783 + 2.41i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.99162388978625479779463245972, −11.13018230071928886783471267727, −10.61895884982588299688223258929, −9.176720200747970496547546509427, −8.737997666994568410471747137826, −6.95294468190062920674623747275, −5.97455290063261675997273944220, −4.11437206136530350261856851586, −2.89402927455475508291492486354, −1.08621226589157458777001933948, 2.46648393126332152477317168277, 4.63195359115671035188846274171, 5.71987552905335270148556211017, 6.83413338908058546733456426659, 7.928937141208519880168401056741, 8.511781394715129384311591969175, 9.773199970662916167778135936181, 10.97217068891730671548765387373, 11.94166541551351640389074579254, 12.99482609168103347836979427984

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