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Properties

Label 2-190-95.49-c1-0-1
Degree $2$
Conductor $190$
Sign $0.994 + 0.103i$
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  + (−0.866 − 0.5i)2-s + (−2.02 − 1.17i)3-s + (0.499 + 0.866i)4-s + (1.07 + 1.96i)5-s + (1.17 + 2.02i)6-s + 1.34i·7-s − 0.999i·8-s + (1.23 + 2.14i)9-s + (0.0529 − 2.23i)10-s + 3.25·11-s − 2.34i·12-s + (1.29 − 0.745i)13-s + (0.670 − 1.16i)14-s + (0.123 − 5.23i)15-s + (−0.5 + 0.866i)16-s + (5.70 + 3.29i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−1.17 − 0.675i)3-s + (0.249 + 0.433i)4-s + (0.479 + 0.877i)5-s + (0.477 + 0.827i)6-s + 0.506i·7-s − 0.353i·8-s + (0.413 + 0.715i)9-s + (0.0167 − 0.706i)10-s + 0.982·11-s − 0.675i·12-s + (0.358 − 0.206i)13-s + (0.179 − 0.310i)14-s + (0.0319 − 1.35i)15-s + (−0.125 + 0.216i)16-s + (1.38 + 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.994 + 0.103i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ 0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702994 - 0.0364601i\)
\(L(\frac12)\) \(\approx\) \(0.702994 - 0.0364601i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-1.07 - 1.96i)T \)
19 \( 1 + (-1.25 + 4.17i)T \)
good3 \( 1 + (2.02 + 1.17i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 1.34iT - 7T^{2} \)
11 \( 1 - 3.25T + 11T^{2} \)
13 \( 1 + (-1.29 + 0.745i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.70 - 3.29i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.86 + 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.65 - 4.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.76T + 31T^{2} \)
37 \( 1 - 5.27iT - 37T^{2} \)
41 \( 1 + (2.79 - 4.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.61 + 3.81i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.806 + 0.465i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.87 + 4.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.0837 - 0.145i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.84 + 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.3 + 6.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.59 - 6.22i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.2 + 7.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.75 - 13.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.313iT - 83T^{2} \)
89 \( 1 + (4.90 + 8.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.4 + 6.01i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.21050389236355941644781557944, −11.53832317149838098341922597276, −10.73843525103919508795076857675, −9.790044125589210025858511252267, −8.583818851546739439998327080182, −7.13280877270769930995964792612, −6.43779140736564151088603401200, −5.44516865958714926559326649733, −3.24641366397674209031039959668, −1.43101387033052295601617334744, 1.09652002898396276908543750617, 4.09675734361949708827388864896, 5.35375828940476653916772014049, 6.05437604588497446257713781834, 7.40538900809474049918971410165, 8.767900741107497723077034942663, 9.767068215924760810067022254939, 10.34106627583508884360191148116, 11.59464196749792122632837001143, 12.15260967285611186720402293259

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