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Properties

Label 2-185-185.174-c1-0-13
Degree $2$
Conductor $185$
Sign $0.852 + 0.521i$
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  + (1.58 − 0.914i)2-s + (1.14 + 0.663i)3-s + (0.673 − 1.16i)4-s + (−1.65 − 1.49i)5-s + 2.42·6-s + (2.49 + 1.44i)7-s + 1.19i·8-s + (−0.618 − 1.07i)9-s + (−4.00 − 0.858i)10-s + 1.49·11-s + (1.54 − 0.894i)12-s + (−5.01 − 2.89i)13-s + 5.27·14-s + (−0.911 − 2.82i)15-s + (2.43 + 4.22i)16-s + (−5.81 + 3.35i)17-s + ⋯
L(s)  = 1  + (1.12 − 0.646i)2-s + (0.663 + 0.383i)3-s + (0.336 − 0.583i)4-s + (−0.741 − 0.670i)5-s + 0.991·6-s + (0.944 + 0.545i)7-s + 0.421i·8-s + (−0.206 − 0.357i)9-s + (−1.26 − 0.271i)10-s + 0.450·11-s + (0.447 − 0.258i)12-s + (−1.39 − 0.802i)13-s + 1.41·14-s + (−0.235 − 0.729i)15-s + (0.609 + 1.05i)16-s + (−1.41 + 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.852 + 0.521i$
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: \((2,\ 185,\ (\ :1/2),\ 0.852 + 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08249 - 0.586552i\)
\(L(\frac12)\) \(\approx\) \(2.08249 - 0.586552i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.65 + 1.49i)T \)
37 \( 1 + (-5.74 + 1.98i)T \)
good2 \( 1 + (-1.58 + 0.914i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.14 - 0.663i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.49 - 1.44i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.49T + 11T^{2} \)
13 \( 1 + (5.01 + 2.89i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.81 - 3.35i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.90 + 3.30i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.31iT - 23T^{2} \)
29 \( 1 + 6.15T + 29T^{2} \)
31 \( 1 - 2.38T + 31T^{2} \)
41 \( 1 + (-3.78 + 6.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 6.09iT - 43T^{2} \)
47 \( 1 - 7.76iT - 47T^{2} \)
53 \( 1 + (0.317 - 0.183i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.43 - 7.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.97 + 6.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.10 - 4.10i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.44 + 2.49i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + (4.49 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.13 + 4.11i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.35 + 4.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.47iT - 97T^{2} \)
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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

−12.49468845222021640518434628349, −11.70859745781563213240248654689, −11.07945600096699681214842103633, −9.386082786565613966770385376253, −8.584237745168462748273072298266, −7.60757405099343854677171450863, −5.55667472068846465639678445245, −4.61920504444179760125731702117, −3.69572619177634886093520243021, −2.32780365289790719584571597439, 2.53226094341939227335801116643, 4.14117507292251987448457657645, 4.88335770563731141949840954451, 6.64685778348670183076116391010, 7.33169549085745039142556351002, 8.160913504772568002161861555541, 9.643837040896547371560212635061, 11.09107334624940447584004129490, 11.84499209234865218636795260341, 13.03104950121607141723850554589

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