Properties

Label 2-380-380.303-c2-0-90
Degree $2$
Conductor $380$
Sign $0.970 + 0.242i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.165 + 1.99i)2-s + (2.36 + 2.36i)3-s + (−3.94 + 0.661i)4-s + (2.79 − 4.14i)5-s + (−4.31 + 5.10i)6-s + (−8.32 − 8.32i)7-s + (−1.97 − 7.75i)8-s + 2.16i·9-s + (8.72 + 4.89i)10-s − 14.7i·11-s + (−10.8 − 7.75i)12-s + (4.62 − 4.62i)13-s + (15.2 − 17.9i)14-s + (16.4 − 3.17i)15-s + (15.1 − 5.21i)16-s + (2.68 − 2.68i)17-s + ⋯
L(s)  = 1  + (0.0829 + 0.996i)2-s + (0.787 + 0.787i)3-s + (−0.986 + 0.165i)4-s + (0.559 − 0.828i)5-s + (−0.719 + 0.850i)6-s + (−1.18 − 1.18i)7-s + (−0.246 − 0.969i)8-s + 0.240i·9-s + (0.872 + 0.489i)10-s − 1.33i·11-s + (−0.906 − 0.646i)12-s + (0.355 − 0.355i)13-s + (1.08 − 1.28i)14-s + (1.09 − 0.211i)15-s + (0.945 − 0.325i)16-s + (0.157 − 0.157i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.970 + 0.242i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.63274 - 0.200693i\)
\(L(\frac12)\) \(\approx\) \(1.63274 - 0.200693i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.165 - 1.99i)T \)
5 \( 1 + (-2.79 + 4.14i)T \)
19 \( 1 + (-12.7 - 14.0i)T \)
good3 \( 1 + (-2.36 - 2.36i)T + 9iT^{2} \)
7 \( 1 + (8.32 + 8.32i)T + 49iT^{2} \)
11 \( 1 + 14.7iT - 121T^{2} \)
13 \( 1 + (-4.62 + 4.62i)T - 169iT^{2} \)
17 \( 1 + (-2.68 + 2.68i)T - 289iT^{2} \)
23 \( 1 + (19.0 - 19.0i)T - 529iT^{2} \)
29 \( 1 + 25.8T + 841T^{2} \)
31 \( 1 + 39.2T + 961T^{2} \)
37 \( 1 + (20.2 + 20.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 2.81iT - 1.68e3T^{2} \)
43 \( 1 + (-32.2 + 32.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-20.7 - 20.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (6.53 - 6.53i)T - 2.80e3iT^{2} \)
59 \( 1 - 14.4iT - 3.48e3T^{2} \)
61 \( 1 - 73.3T + 3.72e3T^{2} \)
67 \( 1 + (-71.1 + 71.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 90.3T + 5.04e3T^{2} \)
73 \( 1 + (30.6 + 30.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 123. iT - 6.24e3T^{2} \)
83 \( 1 + (-75.5 + 75.5i)T - 6.88e3iT^{2} \)
89 \( 1 - 56.2T + 7.92e3T^{2} \)
97 \( 1 + (-99.3 - 99.3i)T + 9.40e3iT^{2} \)
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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

−10.66655683962443992892444348549, −9.707401070754602226250055840885, −9.326440023574259521575835349375, −8.397593315357492487254470252431, −7.45833964860319754032268665187, −6.15369828640048561761931663577, −5.42984080874287514139988930513, −3.78507754666722467190901370172, −3.57969042689162270654688971046, −0.64449672130354951842669254566, 1.98043858921896787756224319652, 2.51321910209238722863871109973, 3.58061861409465550730372922700, 5.31741379912104161192641112763, 6.45869614127870465932135681733, 7.43084406724234727773625545190, 8.743850423386241995158717010379, 9.458419462714553585505123816787, 10.09053458871037248594223144047, 11.25093013298844269475719400151

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