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Properties

Label 2-380-20.7-c1-0-28
Degree $2$
Conductor $380$
Sign $-0.227 + 0.973i$
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  + (0.0365 − 1.41i)2-s + (0.389 + 0.389i)3-s + (−1.99 − 0.103i)4-s + (−1.49 + 1.66i)5-s + (0.564 − 0.536i)6-s + (2.85 − 2.85i)7-s + (−0.218 + 2.81i)8-s − 2.69i·9-s + (2.29 + 2.17i)10-s + 0.969i·11-s + (−0.737 − 0.817i)12-s + (4.39 − 4.39i)13-s + (−3.92 − 4.13i)14-s + (−1.22 + 0.0659i)15-s + (3.97 + 0.412i)16-s + (−2.80 − 2.80i)17-s + ⋯
L(s)  = 1  + (0.0258 − 0.999i)2-s + (0.224 + 0.224i)3-s + (−0.998 − 0.0516i)4-s + (−0.668 + 0.743i)5-s + (0.230 − 0.218i)6-s + (1.07 − 1.07i)7-s + (−0.0774 + 0.996i)8-s − 0.898i·9-s + (0.726 + 0.687i)10-s + 0.292i·11-s + (−0.212 − 0.236i)12-s + (1.21 − 1.21i)13-s + (−1.05 − 1.10i)14-s + (−0.317 + 0.0170i)15-s + (0.994 + 0.103i)16-s + (−0.679 − 0.679i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.227 + 0.973i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.227 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.800953 - 1.01006i\)
\(L(\frac12)\) \(\approx\) \(0.800953 - 1.01006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0365 + 1.41i)T \)
5 \( 1 + (1.49 - 1.66i)T \)
19 \( 1 + T \)
good3 \( 1 + (-0.389 - 0.389i)T + 3iT^{2} \)
7 \( 1 + (-2.85 + 2.85i)T - 7iT^{2} \)
11 \( 1 - 0.969iT - 11T^{2} \)
13 \( 1 + (-4.39 + 4.39i)T - 13iT^{2} \)
17 \( 1 + (2.80 + 2.80i)T + 17iT^{2} \)
23 \( 1 + (-0.648 - 0.648i)T + 23iT^{2} \)
29 \( 1 + 6.30iT - 29T^{2} \)
31 \( 1 - 3.56iT - 31T^{2} \)
37 \( 1 + (-3.34 - 3.34i)T + 37iT^{2} \)
41 \( 1 - 1.73T + 41T^{2} \)
43 \( 1 + (-6.20 - 6.20i)T + 43iT^{2} \)
47 \( 1 + (7.76 - 7.76i)T - 47iT^{2} \)
53 \( 1 + (7.60 - 7.60i)T - 53iT^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + 3.91T + 61T^{2} \)
67 \( 1 + (-0.712 + 0.712i)T - 67iT^{2} \)
71 \( 1 + 5.91iT - 71T^{2} \)
73 \( 1 + (-2.60 + 2.60i)T - 73iT^{2} \)
79 \( 1 + 9.60T + 79T^{2} \)
83 \( 1 + (-10.3 - 10.3i)T + 83iT^{2} \)
89 \( 1 + 2.69iT - 89T^{2} \)
97 \( 1 + (-2.26 - 2.26i)T + 97iT^{2} \)
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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

−11.06386116901268563725666702165, −10.49541792379777118501732557001, −9.487527274116968930673645148158, −8.299317957971838167819198971035, −7.68094431664466921274358702947, −6.28565249856436039390668777994, −4.63925027028159219149161201666, −3.88895646390357997666926205479, −2.89811477854842466622897662898, −0.959379805340540687636079634255, 1.76589473882826348986953562282, 3.96899360859238484586385700023, 4.88515914127580358159311519353, 5.77008658566279578359651320763, 6.99924544474492703888793875004, 8.217542918640059069923666629844, 8.495208152110942636493369286983, 9.160902941142098737607383412301, 10.91486714759167165314623671069, 11.60838486445708952078922610704

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