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Properties

Label 2-187-187.54-c1-0-0
Degree $2$
Conductor $187$
Sign $0.935 + 0.354i$
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  + (−0.926 − 2.23i)2-s + (−1.59 − 0.318i)3-s + (−2.73 + 2.73i)4-s + (1.86 + 2.79i)5-s + (0.770 + 3.87i)6-s + (2.28 + 1.52i)7-s + (4.16 + 1.72i)8-s + (−0.313 − 0.129i)9-s + (4.52 − 6.77i)10-s + (0.278 + 3.30i)11-s + (5.23 − 3.50i)12-s + (−1.29 + 1.29i)13-s + (1.29 − 6.51i)14-s + (−2.10 − 5.07i)15-s − 3.19i·16-s + (−2.13 − 3.52i)17-s + ⋯
L(s)  = 1  + (−0.655 − 1.58i)2-s + (−0.923 − 0.183i)3-s + (−1.36 + 1.36i)4-s + (0.836 + 1.25i)5-s + (0.314 + 1.58i)6-s + (0.862 + 0.576i)7-s + (1.47 + 0.610i)8-s + (−0.104 − 0.0432i)9-s + (1.43 − 2.14i)10-s + (0.0840 + 0.996i)11-s + (1.51 − 1.01i)12-s + (−0.358 + 0.358i)13-s + (0.346 − 1.74i)14-s + (−0.542 − 1.30i)15-s − 0.798i·16-s + (−0.518 − 0.854i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.935 + 0.354i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.935 + 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.636672 - 0.116635i\)
\(L(\frac12)\) \(\approx\) \(0.636672 - 0.116635i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.278 - 3.30i)T \)
17 \( 1 + (2.13 + 3.52i)T \)
good2 \( 1 + (0.926 + 2.23i)T + (-1.41 + 1.41i)T^{2} \)
3 \( 1 + (1.59 + 0.318i)T + (2.77 + 1.14i)T^{2} \)
5 \( 1 + (-1.86 - 2.79i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (-2.28 - 1.52i)T + (2.67 + 6.46i)T^{2} \)
13 \( 1 + (1.29 - 1.29i)T - 13iT^{2} \)
19 \( 1 + (-2.18 - 5.28i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-6.30 + 1.25i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (3.85 + 5.76i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (0.584 - 2.93i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-3.27 - 0.652i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.57 - 1.05i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (2.32 - 5.61i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (-0.752 - 0.752i)T + 47iT^{2} \)
53 \( 1 + (1.56 - 0.648i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.68 + 6.48i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (8.10 - 12.1i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + 4.75iT - 67T^{2} \)
71 \( 1 + (15.1 + 3.00i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-3.38 + 2.25i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (-7.57 + 1.50i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (-2.47 + 1.02i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-2.84 + 2.84i)T - 89iT^{2} \)
97 \( 1 + (-12.4 + 8.34i)T + (37.1 - 89.6i)T^{2} \)
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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

−11.95744181499992562560942090419, −11.50626342866673600144438347883, −10.73246468253950325363252061603, −9.879303367137435355294055837938, −9.047823702287132524891613780697, −7.47316927930317818829848971638, −6.19733642517757647759827540014, −4.82330586428751518406151898018, −2.88826462910328978475302262545, −1.80954227278353719162114834062, 0.863532695390592147271553535782, 4.85081389104433542483328856254, 5.31751648902241099476052987019, 6.23137199250344460351201561014, 7.51133889029790653700035842127, 8.656399754693433193687899290691, 9.162567110991404018498074029323, 10.54549734601739111900189276403, 11.40039752950492064081544963684, 12.98087926986805693741468304562

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