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Properties

Label 2-185-185.103-c1-0-13
Degree $2$
Conductor $185$
Sign $-0.632 + 0.774i$
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  + (−0.437 − 0.757i)2-s + (−0.437 − 1.63i)3-s + (0.616 − 1.06i)4-s + (1.43 − 1.71i)5-s + (−1.04 + 1.04i)6-s + (0.339 + 1.26i)7-s − 2.83·8-s + (0.124 − 0.0716i)9-s + (−1.92 − 0.332i)10-s + 3.57i·11-s + (−2.01 − 0.539i)12-s + (0.404 − 0.700i)13-s + (0.812 − 0.812i)14-s + (−3.43 − 1.58i)15-s + (0.00479 + 0.00830i)16-s + (−3.21 + 1.85i)17-s + ⋯
L(s)  = 1  + (−0.309 − 0.535i)2-s + (−0.252 − 0.942i)3-s + (0.308 − 0.534i)4-s + (0.639 − 0.768i)5-s + (−0.427 + 0.427i)6-s + (0.128 + 0.479i)7-s − 1.00·8-s + (0.0413 − 0.0238i)9-s + (−0.609 − 0.105i)10-s + 1.07i·11-s + (−0.581 − 0.155i)12-s + (0.112 − 0.194i)13-s + (0.217 − 0.217i)14-s + (−0.885 − 0.409i)15-s + (0.00119 + 0.00207i)16-s + (−0.780 + 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.632 + 0.774i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.632 + 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464006 - 0.978618i\)
\(L(\frac12)\) \(\approx\) \(0.464006 - 0.978618i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.43 + 1.71i)T \)
37 \( 1 + (-6.08 + 0.0421i)T \)
good2 \( 1 + (0.437 + 0.757i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.437 + 1.63i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.339 - 1.26i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 3.57iT - 11T^{2} \)
13 \( 1 + (-0.404 + 0.700i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.21 - 1.85i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.17 + 0.315i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.939T + 23T^{2} \)
29 \( 1 + (-0.522 + 0.522i)T - 29iT^{2} \)
31 \( 1 + (-7.77 - 7.77i)T + 31iT^{2} \)
41 \( 1 + (-0.709 - 0.409i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 0.435T + 43T^{2} \)
47 \( 1 + (4.99 - 4.99i)T - 47iT^{2} \)
53 \( 1 + (-1.59 + 5.96i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.62 + 6.05i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (13.0 - 3.50i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-13.0 + 3.49i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.416 - 0.722i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.94 + 2.94i)T - 73iT^{2} \)
79 \( 1 + (9.16 - 2.45i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (1.75 - 6.56i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (14.3 + 3.85i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 1.86iT - 97T^{2} \)
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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

−12.34740521659219071608697537017, −11.44251928846197235369011024313, −10.17928352402362136526296795757, −9.439949310358440859761361299473, −8.344227772438993159699633547664, −6.85713864956382962110659996020, −6.04354370383742547859989002348, −4.81013412277460117218827226561, −2.32523679990221126726326860455, −1.27695578233077420487712342411, 2.83413753071371264286780873586, 4.16587822478373246160820308123, 5.75995880409861877806839794397, 6.72605778404885567875823536152, 7.78848973145225113326057118755, 9.034762188277803394578063043225, 9.964478860128394589551881468055, 10.99188660292802788365055751062, 11.54528540731576398653965972613, 13.19196625711088747137218747482

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