Properties

Label 2-380-380.7-c1-0-19
Degree $2$
Conductor $380$
Sign $0.809 - 0.587i$
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  + (−1.31 + 0.515i)2-s + (0.129 − 0.485i)3-s + (1.46 − 1.35i)4-s + (2.06 − 0.863i)5-s + (0.0789 + 0.705i)6-s + (−2.26 + 2.26i)7-s + (−1.23 + 2.54i)8-s + (2.37 + 1.37i)9-s + (−2.27 + 2.20i)10-s + 1.89i·11-s + (−0.467 − 0.888i)12-s + (0.275 + 1.02i)13-s + (1.81 − 4.14i)14-s + (−0.150 − 1.11i)15-s + (0.311 − 3.98i)16-s + (−0.590 + 2.20i)17-s + ⋯
L(s)  = 1  + (−0.931 + 0.364i)2-s + (0.0750 − 0.280i)3-s + (0.734 − 0.679i)4-s + (0.922 − 0.386i)5-s + (0.0322 + 0.288i)6-s + (−0.855 + 0.855i)7-s + (−0.435 + 0.899i)8-s + (0.793 + 0.457i)9-s + (−0.717 + 0.696i)10-s + 0.572i·11-s + (−0.135 − 0.256i)12-s + (0.0765 + 0.285i)13-s + (0.484 − 1.10i)14-s + (−0.0389 − 0.287i)15-s + (0.0778 − 0.996i)16-s + (−0.143 + 0.534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.809 - 0.587i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.809 - 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00179 + 0.325444i\)
\(L(\frac12)\) \(\approx\) \(1.00179 + 0.325444i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.515i)T \)
5 \( 1 + (-2.06 + 0.863i)T \)
19 \( 1 + (-4.27 + 0.829i)T \)
good3 \( 1 + (-0.129 + 0.485i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.26 - 2.26i)T - 7iT^{2} \)
11 \( 1 - 1.89iT - 11T^{2} \)
13 \( 1 + (-0.275 - 1.02i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.590 - 2.20i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-4.67 + 1.25i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.22 + 1.86i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.974iT - 31T^{2} \)
37 \( 1 + (2.22 + 2.22i)T + 37iT^{2} \)
41 \( 1 + (-5.35 - 9.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.544 + 2.03i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.40 - 8.98i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.980 + 3.66i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.52 + 4.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.56 + 11.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.445 + 1.66i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.55 - 2.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.2 + 3.54i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.65 + 9.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.42 - 9.42i)T + 83iT^{2} \)
89 \( 1 + (6.74 + 3.89i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.47 + 12.9i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.24465946542667170131814889544, −10.15046665290871664737142294898, −9.501555816839297382259227094554, −8.925351682190708056894998083331, −7.72074914912671073917767467186, −6.74829279461436453133046008479, −5.95623924854551640081390451824, −4.89132222065658623549442331134, −2.66593760937327920400502906767, −1.52694002467540805345483930581, 1.09499711561567070618060715390, 2.89592215816260674869757132052, 3.79113380282718055170001445610, 5.66295608667208189423005913348, 6.90508409679892580398798559388, 7.30429945561209902826092094925, 8.926817343150676891427741687978, 9.539753189704382414044104402770, 10.26524897026168056298863478712, 10.82711762957598331594329385545

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