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Properties

Label 2-380-76.31-c1-0-33
Degree $2$
Conductor $380$
Sign $-0.433 + 0.901i$
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  + (0.564 + 1.29i)2-s + (−1.52 − 2.64i)3-s + (−1.36 + 1.46i)4-s + (0.5 + 0.866i)5-s + (2.56 − 3.47i)6-s − 2.58i·7-s + (−2.66 − 0.939i)8-s + (−3.16 + 5.48i)9-s + (−0.840 + 1.13i)10-s − 0.0502i·11-s + (5.95 + 1.36i)12-s + (−2.19 − 1.26i)13-s + (3.34 − 1.45i)14-s + (1.52 − 2.64i)15-s + (−0.288 − 3.98i)16-s + (−3.71 − 6.43i)17-s + ⋯
L(s)  = 1  + (0.399 + 0.916i)2-s + (−0.882 − 1.52i)3-s + (−0.681 + 0.732i)4-s + (0.223 + 0.387i)5-s + (1.04 − 1.41i)6-s − 0.975i·7-s + (−0.943 − 0.332i)8-s + (−1.05 + 1.82i)9-s + (−0.265 + 0.359i)10-s − 0.0151i·11-s + (1.71 + 0.394i)12-s + (−0.608 − 0.351i)13-s + (0.894 − 0.389i)14-s + (0.394 − 0.683i)15-s + (−0.0720 − 0.997i)16-s + (−0.901 − 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.433 + 0.901i$
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: \((2,\ 380,\ (\ :1/2),\ -0.433 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.317156 - 0.504303i\)
\(L(\frac12)\) \(\approx\) \(0.317156 - 0.504303i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.564 - 1.29i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (4.32 + 0.540i)T \)
good3 \( 1 + (1.52 + 2.64i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 2.58iT - 7T^{2} \)
11 \( 1 + 0.0502iT - 11T^{2} \)
13 \( 1 + (2.19 + 1.26i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.71 + 6.43i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.82 + 1.05i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.821 + 0.474i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.30T + 31T^{2} \)
37 \( 1 - 8.51iT - 37T^{2} \)
41 \( 1 + (-9.25 + 5.34i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.12 + 4.11i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.59 + 4.38i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.5 - 6.06i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.90 - 3.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.165 - 0.286i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.82 - 4.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.89 + 6.74i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.13 + 8.89i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.09 - 5.35i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.75iT - 83T^{2} \)
89 \( 1 + (8.19 + 4.72i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.4 - 6.05i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.33765724598925231980628032390, −10.32318757987308528454693973165, −8.887178901575567735176241048369, −7.60402963512964759509475335789, −7.18257649858465942338365015284, −6.48513780163435475754962746738, −5.53091643898032581749034901261, −4.41535870307660939451967270957, −2.52244213337385312186943634370, −0.37056858746339045595551530166, 2.24419118826553353813919420317, 3.91949808166647973226059081062, 4.55440106155882536290394998283, 5.63183843003565510464022082896, 6.15034814429745304766589316948, 8.594951363274726645291855439238, 9.244961944943216959967876906144, 9.984733921479877201051055646126, 10.87690836108170151806000011648, 11.39194359269819591639813228946

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