Properties

Label 2-190-5.4-c1-0-0
Degree $2$
Conductor $190$
Sign $-0.193 - 0.981i$
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  i·2-s + 2.76i·3-s − 4-s + (−2.19 + 0.432i)5-s + 2.76·6-s + 0.761i·7-s + i·8-s − 4.62·9-s + (0.432 + 2.19i)10-s − 0.864·11-s − 2.76i·12-s + 5.62i·13-s + 0.761·14-s + (−1.19 − 6.05i)15-s + 16-s + 3.62i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.59i·3-s − 0.5·4-s + (−0.981 + 0.193i)5-s + 1.12·6-s + 0.287i·7-s + 0.353i·8-s − 1.54·9-s + (0.136 + 0.693i)10-s − 0.260·11-s − 0.797i·12-s + 1.56i·13-s + 0.203·14-s + (−0.308 − 1.56i)15-s + 0.250·16-s + 0.879i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $-0.193 - 0.981i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ -0.193 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.511151 + 0.621707i\)
\(L(\frac12)\) \(\approx\) \(0.511151 + 0.621707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (2.19 - 0.432i)T \)
19 \( 1 + T \)
good3 \( 1 - 2.76iT - 3T^{2} \)
7 \( 1 - 0.761iT - 7T^{2} \)
11 \( 1 + 0.864T + 11T^{2} \)
13 \( 1 - 5.62iT - 13T^{2} \)
17 \( 1 - 3.62iT - 17T^{2} \)
23 \( 1 + 8.01iT - 23T^{2} \)
29 \( 1 - 7.35T + 29T^{2} \)
31 \( 1 - 8.11T + 31T^{2} \)
37 \( 1 + 0.476iT - 37T^{2} \)
41 \( 1 + 2.65T + 41T^{2} \)
43 \( 1 - 6.86iT - 43T^{2} \)
47 \( 1 + 1.25iT - 47T^{2} \)
53 \( 1 - 2.37iT - 53T^{2} \)
59 \( 1 + 4.49T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 1.03iT - 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 16.4iT - 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 + 0.270iT - 83T^{2} \)
89 \( 1 + 0.387T + 89T^{2} \)
97 \( 1 - 8.50iT - 97T^{2} \)
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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

−12.41077713380571947027601922886, −11.67293741349906296720479084870, −10.75375989109082026035379010027, −10.14884691040259805160011263335, −8.966339076742160409515152000891, −8.305784216425290768458823111054, −6.43802464655902387947760643199, −4.61589358589630215739542468215, −4.22455135236498366305667709228, −2.85465039987396600095188827587, 0.74776098080473919257872564369, 3.12557198292951037263457555753, 4.98418674247858018663533438450, 6.24733095303766015821591575478, 7.44196323962049808486462338607, 7.75496749153089660814551954387, 8.715826399655238284055447667960, 10.36118109238719800379241836569, 11.74260368640394501588474954878, 12.37068294356348992738179152336

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