Properties

Label 2-380-95.3-c1-0-4
Degree $2$
Conductor $380$
Sign $0.779 - 0.625i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
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  + (0.432 − 0.927i)3-s + (−0.895 + 2.04i)5-s + (0.864 + 0.231i)7-s + (1.25 + 1.49i)9-s + (−3.29 + 5.70i)11-s + (4.38 − 2.04i)13-s + (1.51 + 1.71i)15-s + (1.76 − 0.154i)17-s + (3.61 − 2.43i)19-s + (0.588 − 0.701i)21-s + (−3.57 + 5.10i)23-s + (−3.39 − 3.67i)25-s + (4.89 − 1.31i)27-s + (3.21 − 2.70i)29-s + (−2.00 + 1.15i)31-s + ⋯
L(s)  = 1  + (0.249 − 0.535i)3-s + (−0.400 + 0.916i)5-s + (0.326 + 0.0875i)7-s + (0.418 + 0.498i)9-s + (−0.992 + 1.71i)11-s + (1.21 − 0.567i)13-s + (0.390 + 0.443i)15-s + (0.427 − 0.0373i)17-s + (0.828 − 0.559i)19-s + (0.128 − 0.153i)21-s + (−0.746 + 1.06i)23-s + (−0.679 − 0.734i)25-s + (0.942 − 0.252i)27-s + (0.597 − 0.501i)29-s + (−0.360 + 0.207i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.779 - 0.625i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.779 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35042 + 0.474842i\)
\(L(\frac12)\) \(\approx\) \(1.35042 + 0.474842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.895 - 2.04i)T \)
19 \( 1 + (-3.61 + 2.43i)T \)
good3 \( 1 + (-0.432 + 0.927i)T + (-1.92 - 2.29i)T^{2} \)
7 \( 1 + (-0.864 - 0.231i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.29 - 5.70i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.38 + 2.04i)T + (8.35 - 9.95i)T^{2} \)
17 \( 1 + (-1.76 + 0.154i)T + (16.7 - 2.95i)T^{2} \)
23 \( 1 + (3.57 - 5.10i)T + (-7.86 - 21.6i)T^{2} \)
29 \( 1 + (-3.21 + 2.70i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.00 - 1.15i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.30 - 3.30i)T + 37iT^{2} \)
41 \( 1 + (2.25 + 6.19i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (2.60 - 1.82i)T + (14.7 - 40.4i)T^{2} \)
47 \( 1 + (-0.488 + 5.57i)T + (-46.2 - 8.16i)T^{2} \)
53 \( 1 + (-9.12 - 6.39i)T + (18.1 + 49.8i)T^{2} \)
59 \( 1 + (1.36 + 1.14i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.13 - 6.45i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (10.8 + 0.953i)T + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (10.2 + 1.81i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 + (-6.56 - 3.06i)T + (46.9 + 55.9i)T^{2} \)
79 \( 1 + (-1.84 + 0.672i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-4.06 + 15.1i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (13.1 + 4.79i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (1.07 + 12.2i)T + (-95.5 + 16.8i)T^{2} \)
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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

−11.49344492907207216718938024035, −10.41792169722250123440960225105, −9.941047598021599558627170621065, −8.359244019181976056555328279017, −7.53758633116939166952438252601, −7.12447347293478041142586256518, −5.69743084343428799990556650356, −4.48214120171272167661528546544, −3.06827100324762539191808788425, −1.83925470326557088506569659379, 1.06034012632355575679399613875, 3.28456210788967161979315601475, 4.14865783482470777022233164787, 5.32019620059466476553895398013, 6.30563952034244197240587838140, 7.931078840675376950550409005069, 8.454210958888261376734813192681, 9.286216749874640283203309370404, 10.38183363635387328482489530736, 11.21465968862413640251645912784

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