Properties

Label 2-380-380.219-c1-0-5
Degree $2$
Conductor $380$
Sign $0.926 - 0.377i$
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  + (−1.33 − 0.479i)2-s + (−1.75 − 2.09i)3-s + (1.54 + 1.27i)4-s + (−1.37 − 1.76i)5-s + (1.33 + 3.62i)6-s + (−1.59 + 2.75i)7-s + (−1.43 − 2.43i)8-s + (−0.772 + 4.38i)9-s + (0.976 + 3.00i)10-s + (−0.876 + 0.506i)11-s + (−0.0361 − 5.45i)12-s + (3.83 + 3.21i)13-s + (3.43 − 2.90i)14-s + (−1.29 + 5.96i)15-s + (0.746 + 3.92i)16-s + (−5.77 + 1.01i)17-s + ⋯
L(s)  = 1  + (−0.940 − 0.338i)2-s + (−1.01 − 1.20i)3-s + (0.770 + 0.637i)4-s + (−0.612 − 0.790i)5-s + (0.543 + 1.47i)6-s + (−0.601 + 1.04i)7-s + (−0.508 − 0.861i)8-s + (−0.257 + 1.46i)9-s + (0.308 + 0.951i)10-s + (−0.264 + 0.152i)11-s + (−0.0104 − 1.57i)12-s + (1.06 + 0.891i)13-s + (0.918 − 0.776i)14-s + (−0.333 + 1.54i)15-s + (0.186 + 0.982i)16-s + (−1.40 + 0.247i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.926 - 0.377i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.926 - 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301479 + 0.0590649i\)
\(L(\frac12)\) \(\approx\) \(0.301479 + 0.0590649i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.479i)T \)
5 \( 1 + (1.37 + 1.76i)T \)
19 \( 1 + (0.956 + 4.25i)T \)
good3 \( 1 + (1.75 + 2.09i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (1.59 - 2.75i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.876 - 0.506i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.83 - 3.21i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (5.77 - 1.01i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-5.67 + 2.06i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-2.84 - 0.501i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.591 - 1.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.25T + 37T^{2} \)
41 \( 1 + (-4.05 - 4.83i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-4.48 - 1.63i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.20 - 6.85i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (7.88 - 2.87i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.555 - 3.15i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (8.86 - 3.22i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-5.98 - 1.05i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (1.39 + 0.509i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (9.08 + 10.8i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (4.30 - 3.60i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.37 - 7.58i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.27 - 3.89i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (-2.56 - 14.5i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.24119064683459996739327971265, −11.07166041765149522111526892457, −9.169504765937515063470475987345, −8.835498946562907661903393351295, −7.70672964854579852117444249198, −6.64921709120709909621521194925, −6.12376980490414518257881596834, −4.55437616145378555815743001315, −2.59907499015622306885056149706, −1.17149481768973565331421304959, 0.36404523114382654048857023226, 3.22993329318444782721397849359, 4.31948112501691537722639824458, 5.75150592038127633047852676171, 6.54078675599745061112067956561, 7.47606646360995412497903153662, 8.627245480445560368851451387643, 9.801236717080866593022560534895, 10.48167508382555810538565939103, 10.96843810120658160381370408531

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