Properties

Label 2-380-380.227-c2-0-33
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: $\chi_{380} (227, \cdot )$
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

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

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