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Properties

Label 2-380-76.31-c1-0-26
Degree $2$
Conductor $380$
Sign $-0.839 + 0.543i$
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  + (−0.293 − 1.38i)2-s + (−0.256 − 0.445i)3-s + (−1.82 + 0.811i)4-s + (0.5 + 0.866i)5-s + (−0.540 + 0.486i)6-s − 1.09i·7-s + (1.65 + 2.29i)8-s + (1.36 − 2.36i)9-s + (1.05 − 0.945i)10-s − 3.04i·11-s + (0.830 + 0.605i)12-s + (−1.26 − 0.732i)13-s + (−1.50 + 0.319i)14-s + (0.256 − 0.445i)15-s + (2.68 − 2.96i)16-s + (−2.28 − 3.95i)17-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.148 − 0.256i)3-s + (−0.913 + 0.405i)4-s + (0.223 + 0.387i)5-s + (−0.220 + 0.198i)6-s − 0.412i·7-s + (0.586 + 0.809i)8-s + (0.455 − 0.789i)9-s + (0.332 − 0.299i)10-s − 0.916i·11-s + (0.239 + 0.174i)12-s + (−0.351 − 0.203i)13-s + (−0.403 + 0.0854i)14-s + (0.0663 − 0.114i)15-s + (0.670 − 0.741i)16-s + (−0.554 − 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.839 + 0.543i$
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: \((2,\ 380,\ (\ :1/2),\ -0.839 + 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.268594 - 0.909640i\)
\(L(\frac12)\) \(\approx\) \(0.268594 - 0.909640i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.293 + 1.38i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (4.20 + 1.15i)T \)
good3 \( 1 + (0.256 + 0.445i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 1.09iT - 7T^{2} \)
11 \( 1 + 3.04iT - 11T^{2} \)
13 \( 1 + (1.26 + 0.732i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.28 + 3.95i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.879 + 0.507i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.38 - 2.53i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.37T + 31T^{2} \)
37 \( 1 + 8.84iT - 37T^{2} \)
41 \( 1 + (-0.478 + 0.276i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.50 - 4.33i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0276 + 0.0159i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.91 - 1.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.46 - 4.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.84 + 3.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.70 + 6.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.153 + 0.266i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.89 + 5.01i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.197 + 0.341i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 + (-12.0 - 6.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.00683 + 0.00394i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.96512286674982477711269485514, −10.22577315514301449505571686684, −9.322853204550988103093299671569, −8.436394991784978396355836275575, −7.23843825707965885107852243234, −6.24960006569958354936502783583, −4.79762353386954563244109264600, −3.64643269811405459457209928255, −2.46562173722060321366639760282, −0.71555475864840549009693430219, 1.93616849820119066282351387525, 4.29098316121497491111763069867, 4.87585120809145480200083807694, 6.05381052863790342299225613954, 6.95752657307189686928191909887, 8.115001127960757152743078991950, 8.739308853463528983108952434960, 10.03750915537966411335906324458, 10.25165237649295958394560314075, 11.81812375746416313184613654511

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