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Properties

Label 2-185-37.26-c1-0-4
Degree $2$
Conductor $185$
Sign $-0.986 - 0.166i$
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  + (1.18 + 2.04i)2-s + (−1.08 + 1.87i)3-s + (−1.79 + 3.10i)4-s + (−0.5 + 0.866i)5-s − 5.12·6-s + (2.11 − 3.65i)7-s − 3.73·8-s + (−0.856 − 1.48i)9-s − 2.36·10-s + 2.37·11-s + (−3.88 − 6.73i)12-s + (−0.945 + 1.63i)13-s + 9.97·14-s + (−1.08 − 1.87i)15-s + (−0.829 − 1.43i)16-s + (−2.62 − 4.55i)17-s + ⋯
L(s)  = 1  + (0.835 + 1.44i)2-s + (−0.626 + 1.08i)3-s + (−0.895 + 1.55i)4-s + (−0.223 + 0.387i)5-s − 2.09·6-s + (0.797 − 1.38i)7-s − 1.32·8-s + (−0.285 − 0.494i)9-s − 0.747·10-s + 0.717·11-s + (−1.12 − 1.94i)12-s + (−0.262 + 0.454i)13-s + 2.66·14-s + (−0.280 − 0.485i)15-s + (−0.207 − 0.359i)16-s + (−0.637 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.986 - 0.166i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.986 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121463 + 1.45024i\)
\(L(\frac12)\) \(\approx\) \(0.121463 + 1.45024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (2.93 + 5.32i)T \)
good2 \( 1 + (-1.18 - 2.04i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.08 - 1.87i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.11 + 3.65i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.37T + 11T^{2} \)
13 \( 1 + (0.945 - 1.63i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.62 + 4.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.56 - 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.56T + 23T^{2} \)
29 \( 1 - 7.70T + 29T^{2} \)
31 \( 1 + 2.35T + 31T^{2} \)
41 \( 1 + (-5.15 + 8.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 0.629T + 43T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 + (3.15 + 5.45i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.47 - 11.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.536 - 0.930i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.17 - 2.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.34 - 4.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.96T + 73T^{2} \)
79 \( 1 + (-1.63 + 2.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.62 + 11.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.941 + 1.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.74T + 97T^{2} \)
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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

−13.62295035342351143150156568806, −12.12629490697376552869476178401, −11.02797208664090235720130456178, −10.30312655173954631744938815315, −8.849114628933166737905168950170, −7.47059120518521746777708940844, −6.85444289491577395082225862721, −5.51917283851456259758256055724, −4.38366851813715993409355529057, −4.03851618384197044817544221755, 1.34749033274230831131606979363, 2.55023916946727840660720838719, 4.41444883663791721149649272189, 5.44242822634579733680125714996, 6.54060940450842835311455554386, 8.231047300965474461501955808840, 9.257089708038593652116386119687, 10.83489206818838222130225965023, 11.50424290624327314058408774921, 12.27997705207412884806583378302

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