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Properties

Label 2-190-5.4-c1-0-1
Degree $2$
Conductor $190$
Sign $-0.948 - 0.316i$
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 + 2.41i·3-s − 4-s + (−0.707 + 2.12i)5-s − 2.41·6-s − 1.58i·7-s − i·8-s − 2.82·9-s + (−2.12 − 0.707i)10-s + 1.41·11-s − 2.41i·12-s + 0.171i·13-s + 1.58·14-s + (−5.12 − 1.70i)15-s + 16-s − i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.39i·3-s − 0.5·4-s + (−0.316 + 0.948i)5-s − 0.985·6-s − 0.599i·7-s − 0.353i·8-s − 0.942·9-s + (−0.670 − 0.223i)10-s + 0.426·11-s − 0.696i·12-s + 0.0475i·13-s + 0.423·14-s + (−1.32 − 0.440i)15-s + 0.250·16-s − 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $-0.948 - 0.316i$
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: \((2,\ 190,\ (\ :1/2),\ -0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165606 + 1.02051i\)
\(L(\frac12)\) \(\approx\) \(0.165606 + 1.02051i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 2.12i)T \)
19 \( 1 - T \)
good3 \( 1 - 2.41iT - 3T^{2} \)
7 \( 1 + 1.58iT - 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 - 0.171iT - 13T^{2} \)
17 \( 1 + iT - 17T^{2} \)
23 \( 1 - 9.24iT - 23T^{2} \)
29 \( 1 - 5.82T + 29T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 + 8.48iT - 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 5.75T + 61T^{2} \)
67 \( 1 + 13.2iT - 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + 5.48iT - 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 2.48iT - 83T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 - 11.6iT - 97T^{2} \)
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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

−13.34346586336105038487837998677, −11.69604908165872583763493940479, −10.84901936046373798613611491902, −9.944956264859400231546112847225, −9.226144204304388266739620464857, −7.79979579212299474698012779559, −6.86047977285913236053756656268, −5.52828841874755400543496406976, −4.23857975780464608175410516167, −3.40229962401988518157623825290, 1.02863569598538131335492315274, 2.47081895956652662150183068552, 4.34788560192156619957240912078, 5.75494901080164014983248105951, 7.00122589752917130978013107794, 8.331790816653528889246242339463, 8.795189014303045954545592437461, 10.22131867102649119884809628737, 11.65936551462715246284612406980, 12.21325008976842428965606723517

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