Properties

Label 2-190-19.4-c1-0-5
Degree $2$
Conductor $190$
Sign $0.153 - 0.988i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
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  + (0.939 + 0.342i)2-s + (0.552 + 3.13i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)5-s + (−0.552 + 3.13i)6-s + (1.67 − 2.89i)7-s + (0.500 + 0.866i)8-s + (−6.68 + 2.43i)9-s + (0.939 − 0.342i)10-s + (−3.09 − 5.35i)11-s + (−1.58 + 2.75i)12-s + (−0.128 + 0.727i)13-s + (2.56 − 2.15i)14-s + (2.43 + 2.04i)15-s + (0.173 + 0.984i)16-s + (−0.815 − 0.296i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.318 + 1.80i)3-s + (0.383 + 0.321i)4-s + (0.342 − 0.287i)5-s + (−0.225 + 1.27i)6-s + (0.632 − 1.09i)7-s + (0.176 + 0.306i)8-s + (−2.22 + 0.810i)9-s + (0.297 − 0.108i)10-s + (−0.932 − 1.61i)11-s + (−0.458 + 0.794i)12-s + (−0.0355 + 0.201i)13-s + (0.685 − 0.574i)14-s + (0.628 + 0.527i)15-s + (0.0434 + 0.246i)16-s + (−0.197 − 0.0720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.153 - 0.988i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ 0.153 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41253 + 1.20989i\)
\(L(\frac12)\) \(\approx\) \(1.41253 + 1.20989i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (2.59 - 3.49i)T \)
good3 \( 1 + (-0.552 - 3.13i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-1.67 + 2.89i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.09 + 5.35i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.128 - 0.727i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.815 + 0.296i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (0.875 + 0.735i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-7.32 + 2.66i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.23 - 5.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.68T + 37T^{2} \)
41 \( 1 + (-1.68 - 9.54i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.42 + 1.19i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (0.311 - 0.113i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (5.44 + 4.56i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (4.56 + 1.66i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-5.53 - 4.64i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.74 - 1.00i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (5.27 - 4.42i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.35 - 7.69i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.399 + 2.26i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.72 + 9.91i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.404 + 2.29i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-6.43 - 2.34i)T + (74.3 + 62.3i)T^{2} \)
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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

−13.14094628232349220847130900726, −11.46450012838157336785502908999, −10.70332950116476708161736982484, −10.11573266787273703488368899881, −8.682711126322784627657448961145, −8.004035979548521735804470869964, −6.06205267520292137297674257416, −4.99424262789684390254347485610, −4.17909346323554715262651451633, −3.04637925271467669850723554427, 2.01332324628375114090870676913, 2.56746325981405006283886389191, 4.98757254420053467030023846200, 6.09857470540345549732749629309, 7.12274249735815805489234003884, 7.973551088283091595785606838706, 9.161942313549083141985882675111, 10.70399584746507087867410048972, 11.85470053471713362043538908840, 12.51729227603227476382467688435

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