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Properties

Label 2-185-185.174-c1-0-11
Degree $2$
Conductor $185$
Sign $-0.0177 + 0.999i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
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  + (−1.85 + 1.07i)2-s + (0.858 + 0.495i)3-s + (1.29 − 2.24i)4-s + (−1.59 − 1.56i)5-s − 2.12·6-s + (−1.20 − 0.693i)7-s + 1.25i·8-s + (−1.00 − 1.74i)9-s + (4.63 + 1.20i)10-s − 6.34·11-s + (2.22 − 1.28i)12-s + (−0.416 − 0.240i)13-s + 2.96·14-s + (−0.590 − 2.13i)15-s + (1.23 + 2.14i)16-s + (0.983 − 0.567i)17-s + ⋯
L(s)  = 1  + (−1.31 + 0.757i)2-s + (0.495 + 0.286i)3-s + (0.646 − 1.12i)4-s + (−0.712 − 0.701i)5-s − 0.867·6-s + (−0.453 − 0.262i)7-s + 0.445i·8-s + (−0.336 − 0.582i)9-s + (1.46 + 0.380i)10-s − 1.91·11-s + (0.641 − 0.370i)12-s + (−0.115 − 0.0667i)13-s + 0.793·14-s + (−0.152 − 0.551i)15-s + (0.309 + 0.536i)16-s + (0.238 − 0.137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0177 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0177 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.0177 + 0.999i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.0177 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.186366 - 0.189706i\)
\(L(\frac12)\) \(\approx\) \(0.186366 - 0.189706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.59 + 1.56i)T \)
37 \( 1 + (-4.79 - 3.74i)T \)
good2 \( 1 + (1.85 - 1.07i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.858 - 0.495i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.20 + 0.693i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 6.34T + 11T^{2} \)
13 \( 1 + (0.416 + 0.240i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.983 + 0.567i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.17 + 2.03i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.42iT - 23T^{2} \)
29 \( 1 - 4.51T + 29T^{2} \)
31 \( 1 + 4.97T + 31T^{2} \)
41 \( 1 + (3.53 - 6.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 0.683iT - 43T^{2} \)
47 \( 1 - 7.99iT - 47T^{2} \)
53 \( 1 + (6.99 - 4.03i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.02 + 8.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.08 + 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.31 + 4.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.656 - 1.13i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.46iT - 73T^{2} \)
79 \( 1 + (-0.205 + 0.355i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-14.1 + 8.15i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.79 - 3.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.68iT - 97T^{2} \)
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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

−12.44326194399279469563096615145, −10.98786413792970751210869556368, −9.969428700324455653557915557791, −9.190661731186881272748550807676, −8.195203568473269117247757797933, −7.72640898637979929885781539263, −6.40005210369230027666019909598, −4.87391837892222851863783573466, −3.15669690753991002328318203817, −0.32277410012044723088176127649, 2.33683310346234184431647846816, 3.19791270963639876263768075645, 5.45132636998222017402729370582, 7.41688894856694176220274794202, 7.86792031633868060611505632135, 8.792173900436784831886649237903, 10.07971342024619023562557864312, 10.66099146521347209905324930737, 11.55767599068988699600290988815, 12.60401669346221801774774479492

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