Properties

Label 2-380-380.227-c2-0-92
Degree $2$
Conductor $380$
Sign $0.0869 + 0.996i$
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  + (−1.77 + 0.912i)2-s + (3.66 − 3.66i)3-s + (2.33 − 3.24i)4-s + (4.37 − 2.42i)5-s + (−3.18 + 9.87i)6-s + (−2.51 + 2.51i)7-s + (−1.18 + 7.91i)8-s − 17.9i·9-s + (−5.56 + 8.30i)10-s − 14.1i·11-s + (−3.35 − 20.4i)12-s + (−0.192 − 0.192i)13-s + (2.18 − 6.77i)14-s + (7.14 − 24.9i)15-s + (−5.10 − 15.1i)16-s + (18.2 + 18.2i)17-s + ⋯
L(s)  = 1  + (−0.889 + 0.456i)2-s + (1.22 − 1.22i)3-s + (0.583 − 0.812i)4-s + (0.874 − 0.484i)5-s + (−0.530 + 1.64i)6-s + (−0.359 + 0.359i)7-s + (−0.148 + 0.988i)8-s − 1.99i·9-s + (−0.556 + 0.830i)10-s − 1.28i·11-s + (−0.279 − 1.70i)12-s + (−0.0148 − 0.0148i)13-s + (0.155 − 0.484i)14-s + (0.476 − 1.66i)15-s + (−0.319 − 0.947i)16-s + (1.07 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0869 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0869 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0869 + 0.996i$
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: \((2,\ 380,\ (\ :1),\ 0.0869 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.38367 - 1.26815i\)
\(L(\frac12)\) \(\approx\) \(1.38367 - 1.26815i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.77 - 0.912i)T \)
5 \( 1 + (-4.37 + 2.42i)T \)
19 \( 1 + (-17.8 - 6.50i)T \)
good3 \( 1 + (-3.66 + 3.66i)T - 9iT^{2} \)
7 \( 1 + (2.51 - 2.51i)T - 49iT^{2} \)
11 \( 1 + 14.1iT - 121T^{2} \)
13 \( 1 + (0.192 + 0.192i)T + 169iT^{2} \)
17 \( 1 + (-18.2 - 18.2i)T + 289iT^{2} \)
23 \( 1 + (18.5 + 18.5i)T + 529iT^{2} \)
29 \( 1 + 31.6T + 841T^{2} \)
31 \( 1 + 35.7T + 961T^{2} \)
37 \( 1 + (-17.7 + 17.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 49.7iT - 1.68e3T^{2} \)
43 \( 1 + (-13.5 - 13.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-37.4 + 37.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (-20.7 - 20.7i)T + 2.80e3iT^{2} \)
59 \( 1 - 17.7iT - 3.48e3T^{2} \)
61 \( 1 + 85.5T + 3.72e3T^{2} \)
67 \( 1 + (28.1 + 28.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 62.1T + 5.04e3T^{2} \)
73 \( 1 + (24.6 - 24.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 116. iT - 6.24e3T^{2} \)
83 \( 1 + (-26.7 - 26.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 149.T + 7.92e3T^{2} \)
97 \( 1 + (9.47 - 9.47i)T - 9.40e3iT^{2} \)
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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

−10.62632345152574618581149437042, −9.522912142076158832665053953481, −8.949469443386674436192966517420, −8.158330047394876174473263029107, −7.50714341490038578248660221319, −6.13354145991597092304473961671, −5.80003650511597969761167982598, −3.25738590100679038312297512333, −2.05144259604921353791019434587, −1.00934590326918113858963445506, 1.95430908492449396207820225362, 3.00835904991001215794947808984, 3.88007993497697975286180688256, 5.36601376881062458363410355370, 7.18934519642963395170361306684, 7.68274550994441226552186999844, 9.239012015837530751775970641099, 9.496948721428959427504895415876, 10.06247525146908755967203432718, 10.83652157567221295610619318325

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