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Properties

Label 2-380-19.4-c1-0-5
Degree $2$
Conductor $380$
Sign $-0.633 + 0.773i$
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  + (−0.312 − 1.77i)3-s + (−0.766 + 0.642i)5-s + (−0.336 + 0.583i)7-s + (−0.222 + 0.0811i)9-s + (−3.21 − 5.56i)11-s + (0.791 − 4.48i)13-s + (1.37 + 1.15i)15-s + (1.21 + 0.442i)17-s + (−4.35 + 0.223i)19-s + (1.13 + 0.414i)21-s + (−1.04 − 0.874i)23-s + (0.173 − 0.984i)25-s + (−2.48 − 4.30i)27-s + (−6.80 + 2.47i)29-s + (−2.97 + 5.14i)31-s + ⋯
L(s)  = 1  + (−0.180 − 1.02i)3-s + (−0.342 + 0.287i)5-s + (−0.127 + 0.220i)7-s + (−0.0742 + 0.0270i)9-s + (−0.968 − 1.67i)11-s + (0.219 − 1.24i)13-s + (0.355 + 0.298i)15-s + (0.294 + 0.107i)17-s + (−0.998 + 0.0513i)19-s + (0.248 + 0.0904i)21-s + (−0.217 − 0.182i)23-s + (0.0347 − 0.196i)25-s + (−0.478 − 0.828i)27-s + (−1.26 + 0.459i)29-s + (−0.533 + 0.924i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.633 + 0.773i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.633 + 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396676 - 0.837053i\)
\(L(\frac12)\) \(\approx\) \(0.396676 - 0.837053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (4.35 - 0.223i)T \)
good3 \( 1 + (0.312 + 1.77i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (0.336 - 0.583i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.21 + 5.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.791 + 4.48i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.21 - 0.442i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (1.04 + 0.874i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (6.80 - 2.47i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.97 - 5.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 + (0.463 + 2.62i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-7.31 + 6.14i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-11.0 + 4.01i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (2.74 + 2.30i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.150 - 0.0547i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-5.54 - 4.65i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (5.13 - 1.86i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-9.00 + 7.55i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.643 + 3.64i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.11 + 6.32i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.52 - 4.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.26 + 7.14i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-7.40 - 2.69i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.90711151052055026368385706618, −10.52337117470567579094885808303, −8.930039962265394898868993738376, −8.024017999825462152246093411565, −7.41094021857774148315798076224, −6.13154582196715886124357718074, −5.56342606183808125271156154473, −3.71257856929812194318027072545, −2.53438661183816320763360472389, −0.62502849430948741168328928531, 2.17661740608818923412612637734, 4.17422313958964458001267519820, 4.40289331846486425365774855691, 5.69601254069784910327157804060, 7.11698681350547132033547892807, 7.88625357117345527820507613529, 9.370907773676584139559722234585, 9.698974954853141514575128882910, 10.74765740050002640834865020788, 11.45669413732553355520554373255

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