Properties

Label 2-185-185.84-c1-0-12
Degree $2$
Conductor $185$
Sign $0.838 + 0.544i$
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  + (0.153 + 0.0887i)2-s + (1.48 − 0.854i)3-s + (−0.984 − 1.70i)4-s + (1.36 + 1.77i)5-s + 0.303·6-s + (1.21 − 0.703i)7-s − 0.704i·8-s + (−0.0386 + 0.0668i)9-s + (0.0519 + 0.393i)10-s − 2.99·11-s + (−2.91 − 1.68i)12-s + (5.66 − 3.27i)13-s + 0.250·14-s + (3.53 + 1.46i)15-s + (−1.90 + 3.30i)16-s + (−5.35 − 3.09i)17-s + ⋯
L(s)  = 1  + (0.108 + 0.0627i)2-s + (0.854 − 0.493i)3-s + (−0.492 − 0.852i)4-s + (0.609 + 0.793i)5-s + 0.123·6-s + (0.460 − 0.266i)7-s − 0.249i·8-s + (−0.0128 + 0.0222i)9-s + (0.0164 + 0.124i)10-s − 0.902·11-s + (−0.841 − 0.485i)12-s + (1.57 − 0.907i)13-s + 0.0668·14-s + (0.912 + 0.377i)15-s + (−0.476 + 0.825i)16-s + (−1.29 − 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.838 + 0.544i$
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: \((2,\ 185,\ (\ :1/2),\ 0.838 + 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48587 - 0.440193i\)
\(L(\frac12)\) \(\approx\) \(1.48587 - 0.440193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.36 - 1.77i)T \)
37 \( 1 + (5.97 - 1.16i)T \)
good2 \( 1 + (-0.153 - 0.0887i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.48 + 0.854i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.21 + 0.703i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.99T + 11T^{2} \)
13 \( 1 + (-5.66 + 3.27i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.35 + 3.09i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.790 - 1.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.66iT - 23T^{2} \)
29 \( 1 + 5.85T + 29T^{2} \)
31 \( 1 - 7.52T + 31T^{2} \)
41 \( 1 + (-0.445 - 0.771i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 0.00149iT - 43T^{2} \)
47 \( 1 + 4.55iT - 47T^{2} \)
53 \( 1 + (1.72 + 0.997i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.47 - 4.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.74 + 8.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.04 + 1.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.0847 + 0.146i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.7iT - 73T^{2} \)
79 \( 1 + (-3.12 - 5.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.13 - 4.69i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.69 + 11.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.77iT - 97T^{2} \)
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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

−13.27403893830106975050555468237, −11.14606987264594703969749486063, −10.63200462005664861947250587028, −9.492703743583530311421154686339, −8.452890647183995326414841658937, −7.45480934679318941969979381363, −6.14750651891320098543514828478, −5.13083010612616416878345540021, −3.29007578078732272455749843947, −1.79254808765929427130656588095, 2.35060363276887788516302095394, 3.89779864585728076193238433605, 4.79100111939060648790182264137, 6.33079217816680788180694842089, 8.213503985277720852210210430948, 8.663938895583183110835590609911, 9.271074596641965309349769340453, 10.68286204275233120913642092773, 11.85614890811945717682382633523, 13.01268068744886036808913854074

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