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Properties

Label 2-380-95.13-c1-0-5
Degree $2$
Conductor $380$
Sign $0.0247 + 0.999i$
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  + (−1.41 + 0.660i)3-s + (−0.949 + 2.02i)5-s + (−1.12 − 4.18i)7-s + (−0.359 + 0.428i)9-s + (−0.627 − 1.08i)11-s + (1.92 − 4.12i)13-s + (0.00842 − 3.49i)15-s + (0.0727 − 0.831i)17-s + (3.83 − 2.06i)19-s + (4.35 + 5.18i)21-s + (0.587 − 0.411i)23-s + (−3.19 − 3.84i)25-s + (1.43 − 5.37i)27-s + (−3.22 − 2.70i)29-s + (−6.47 − 3.74i)31-s + ⋯
L(s)  = 1  + (−0.817 + 0.381i)3-s + (−0.424 + 0.905i)5-s + (−0.423 − 1.58i)7-s + (−0.119 + 0.142i)9-s + (−0.189 − 0.327i)11-s + (0.533 − 1.14i)13-s + (0.00217 − 0.902i)15-s + (0.0176 − 0.201i)17-s + (0.880 − 0.474i)19-s + (0.949 + 1.13i)21-s + (0.122 − 0.0858i)23-s + (−0.639 − 0.769i)25-s + (0.276 − 1.03i)27-s + (−0.599 − 0.503i)29-s + (−1.16 − 0.671i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0247 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0247 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0247 + 0.999i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.0247 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.406824 - 0.396869i\)
\(L(\frac12)\) \(\approx\) \(0.406824 - 0.396869i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.949 - 2.02i)T \)
19 \( 1 + (-3.83 + 2.06i)T \)
good3 \( 1 + (1.41 - 0.660i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (1.12 + 4.18i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.627 + 1.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.92 + 4.12i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (-0.0727 + 0.831i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (-0.587 + 0.411i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (3.22 + 2.70i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (6.47 + 3.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.31 - 1.31i)T + 37iT^{2} \)
41 \( 1 + (2.83 - 7.79i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-6.72 + 9.60i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (9.51 - 0.832i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (5.49 + 7.84i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (2.93 - 2.46i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.31 - 7.44i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.616 - 7.04i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (-11.6 + 2.05i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (-1.19 - 2.56i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (5.16 + 1.88i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-7.22 + 1.93i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-3.31 + 1.20i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (5.44 + 0.476i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.02332731720094441070237244725, −10.47011898415342794997875118100, −9.720807911167160136962838485458, −8.034328281869486620205855763364, −7.34876647548678127255377119536, −6.34495174473809424184780666431, −5.31581392332424626603938860914, −4.00684613659003188142131947993, −3.08639896389127500819604279511, −0.42494364535310490103427595573, 1.66735834279711901428139891436, 3.48038157984662997749987226289, 5.02472508342387490630440568841, 5.73574965020482228406852108820, 6.62992482882776763498971102651, 7.897045709866135390167557081913, 9.122104888749233139716004157764, 9.292819316786769734049419746483, 11.07263057681717519689608476021, 11.71001379684195367456355631750

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