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Properties

Label 2-380-76.31-c1-0-31
Degree $2$
Conductor $380$
Sign $0.217 + 0.976i$
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  + (0.759 − 1.19i)2-s + (0.600 + 1.04i)3-s + (−0.847 − 1.81i)4-s + (0.5 + 0.866i)5-s + (1.69 + 0.0729i)6-s − 4.07i·7-s + (−2.80 − 0.363i)8-s + (0.777 − 1.34i)9-s + (1.41 + 0.0606i)10-s + 0.526i·11-s + (1.37 − 1.97i)12-s + (4.15 + 2.39i)13-s + (−4.86 − 3.09i)14-s + (−0.600 + 1.04i)15-s + (−2.56 + 3.07i)16-s + (−2.96 − 5.13i)17-s + ⋯
L(s)  = 1  + (0.536 − 0.843i)2-s + (0.346 + 0.600i)3-s + (−0.423 − 0.905i)4-s + (0.223 + 0.387i)5-s + (0.693 + 0.0297i)6-s − 1.54i·7-s + (−0.991 − 0.128i)8-s + (0.259 − 0.449i)9-s + (0.446 + 0.0191i)10-s + 0.158i·11-s + (0.397 − 0.568i)12-s + (1.15 + 0.664i)13-s + (−1.29 − 0.826i)14-s + (−0.155 + 0.268i)15-s + (−0.640 + 0.767i)16-s + (−0.719 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.217 + 0.976i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.217 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53812 - 1.23365i\)
\(L(\frac12)\) \(\approx\) \(1.53812 - 1.23365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.759 + 1.19i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-3.48 - 2.61i)T \)
good3 \( 1 + (-0.600 - 1.04i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 4.07iT - 7T^{2} \)
11 \( 1 - 0.526iT - 11T^{2} \)
13 \( 1 + (-4.15 - 2.39i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.96 + 5.13i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.08 - 0.628i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.69 + 1.55i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.0294T + 31T^{2} \)
37 \( 1 - 6.86iT - 37T^{2} \)
41 \( 1 + (6.78 - 3.91i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.46 - 2.57i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.907 + 0.523i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.74 - 1.58i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.03 - 1.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.47 - 9.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.00 - 1.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.94 - 5.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.45 + 2.51i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.81 - 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.72iT - 83T^{2} \)
89 \( 1 + (8.63 + 4.98i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.0 + 7.51i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.17568128740956861807537149762, −10.18987268304971085907715573643, −9.747531837127710603928419139182, −8.770888248833561033094172248314, −7.18716345116146918680226260532, −6.34285792202955168745738080155, −4.79346405013206650924185353583, −3.94489049031322302262286652909, −3.16402400634924360763339305733, −1.28462582184765067789697518568, 2.08491132545286789693196150700, 3.44498235180054033097504406868, 5.01441150071383534629081826608, 5.78231006933712625865199046238, 6.67075292590715656146091712097, 7.924068656112196634716315887118, 8.589280133731456751093602537605, 9.161823015115017773947959145634, 10.78186917092677467254033544096, 11.92242277754796785648324519078

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