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Properties

Label 2-185-37.10-c1-0-3
Degree $2$
Conductor $185$
Sign $0.965 - 0.259i$
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  + (−0.155 + 0.268i)2-s + (−1.03 − 1.78i)3-s + (0.951 + 1.64i)4-s + (0.5 + 0.866i)5-s + 0.639·6-s + (1.06 + 1.83i)7-s − 1.21·8-s + (−0.625 + 1.08i)9-s − 0.310·10-s + 5.48·11-s + (1.96 − 3.39i)12-s + (−0.826 − 1.43i)13-s − 0.657·14-s + (1.03 − 1.78i)15-s + (−1.71 + 2.97i)16-s + (1.42 − 2.46i)17-s + ⋯
L(s)  = 1  + (−0.109 + 0.189i)2-s + (−0.595 − 1.03i)3-s + (0.475 + 0.824i)4-s + (0.223 + 0.387i)5-s + 0.261·6-s + (0.400 + 0.694i)7-s − 0.428·8-s + (−0.208 + 0.361i)9-s − 0.0981·10-s + 1.65·11-s + (0.566 − 0.981i)12-s + (−0.229 − 0.396i)13-s − 0.175·14-s + (0.266 − 0.461i)15-s + (−0.428 + 0.742i)16-s + (0.344 − 0.597i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.965 - 0.259i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12327 + 0.148389i\)
\(L(\frac12)\) \(\approx\) \(1.12327 + 0.148389i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 - 5.58i)T \)
good2 \( 1 + (0.155 - 0.268i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.03 + 1.78i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.06 - 1.83i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.48T + 11T^{2} \)
13 \( 1 + (0.826 + 1.43i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.42 + 2.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.19 - 2.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.29T + 23T^{2} \)
29 \( 1 + 0.459T + 29T^{2} \)
31 \( 1 + 7.00T + 31T^{2} \)
41 \( 1 + (3.63 + 6.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4.39T + 43T^{2} \)
47 \( 1 + 0.310T + 47T^{2} \)
53 \( 1 + (-6.00 + 10.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.14 - 7.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.10 + 8.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.664 + 1.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.293 + 0.508i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 + (-1.41 - 2.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.91 - 10.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.40 + 14.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 18.5T + 97T^{2} \)
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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

−12.22727892360312240857312443559, −11.98217125742242793634081288702, −11.13401617987251934545524998816, −9.499395389577307890779286938186, −8.394751528940216652260682818390, −7.22616743952558598523282542911, −6.63884955838643196243366614048, −5.55742530962537891250011509433, −3.50268332517667265698659000127, −1.82700343890040531540529183737, 1.47485728759692458017305214418, 3.93234909917639854572488259697, 4.96907037415115289276646352299, 6.05703275672667457704431400639, 7.20389077172651188661137735711, 9.047212366797795936077329056675, 9.705536567431877452660555734566, 10.67765370289685005900996428547, 11.29212735096118860040485244384, 12.18511478595230186620024437472

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