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Properties

Label 2-185-185.84-c1-0-14
Degree $2$
Conductor $185$
Sign $0.837 - 0.545i$
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  + (1.99 + 1.15i)2-s + (1.81 − 1.04i)3-s + (1.66 + 2.88i)4-s + (−1.88 − 1.19i)5-s + 4.82·6-s + (−3.28 + 1.89i)7-s + 3.06i·8-s + (0.687 − 1.18i)9-s + (−2.39 − 4.56i)10-s − 3.98·11-s + (6.02 + 3.47i)12-s + (5.14 − 2.96i)13-s − 8.75·14-s + (−4.67 − 0.188i)15-s + (−0.206 + 0.358i)16-s + (2.60 + 1.50i)17-s + ⋯
L(s)  = 1  + (1.41 + 0.815i)2-s + (1.04 − 0.603i)3-s + (0.831 + 1.44i)4-s + (−0.845 − 0.534i)5-s + 1.97·6-s + (−1.24 + 0.717i)7-s + 1.08i·8-s + (0.229 − 0.396i)9-s + (−0.758 − 1.44i)10-s − 1.20·11-s + (1.73 + 1.00i)12-s + (1.42 − 0.823i)13-s − 2.34·14-s + (−1.20 − 0.0487i)15-s + (−0.0516 + 0.0895i)16-s + (0.630 + 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.837 - 0.545i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.837 - 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40566 + 0.714388i\)
\(L(\frac12)\) \(\approx\) \(2.40566 + 0.714388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.88 + 1.19i)T \)
37 \( 1 + (-3.31 + 5.09i)T \)
good2 \( 1 + (-1.99 - 1.15i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.81 + 1.04i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (3.28 - 1.89i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.98T + 11T^{2} \)
13 \( 1 + (-5.14 + 2.96i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.60 - 1.50i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.477 + 0.826i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.297iT - 23T^{2} \)
29 \( 1 + 6.45T + 29T^{2} \)
31 \( 1 + 1.69T + 31T^{2} \)
41 \( 1 + (-3.89 - 6.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 4.65iT - 43T^{2} \)
47 \( 1 - 3.36iT - 47T^{2} \)
53 \( 1 + (1.67 + 0.965i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.90 + 5.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.95 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.15 + 4.70i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.41 + 11.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.763iT - 73T^{2} \)
79 \( 1 + (-4.90 - 8.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (13.4 + 7.77i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.32 - 4.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.16iT - 97T^{2} \)
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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

−12.84753797540493896295908721691, −12.65177733764239126930231363150, −11.08896874892104383047825881307, −9.313324985707380159580960315503, −8.109333103531287332640221587581, −7.65530092595742716687194031030, −6.24102297825339212565124899321, −5.32720913289655971455541296084, −3.65202439255841853168477778878, −2.94843467925956770235755666449, 2.77007105122212471167256027792, 3.60050813463324833490997755462, 4.13840363542645427933160063431, 5.89514274052428356644621415434, 7.23131108970342832132117457099, 8.578652030468600136861578648495, 9.939908520596706162476901542908, 10.68134402185995506825972538486, 11.59843815161724631006056980673, 12.77774772820576233574038832190

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