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Properties

Label 2-380-95.33-c1-0-5
Degree $2$
Conductor $380$
Sign $0.194 + 0.980i$
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  + (−1.61 − 2.30i)3-s + (1.99 − 1.01i)5-s + (4.91 + 1.31i)7-s + (−1.67 + 4.59i)9-s + (1.71 − 2.96i)11-s + (1.74 + 1.22i)13-s + (−5.54 − 2.94i)15-s + (−2.63 + 5.65i)17-s + (1.56 − 4.07i)19-s + (−4.88 − 13.4i)21-s + (−2.25 − 0.197i)23-s + (2.93 − 4.04i)25-s + (5.13 − 1.37i)27-s + (−5.51 − 2.00i)29-s + (−0.923 + 0.533i)31-s + ⋯
L(s)  = 1  + (−0.930 − 1.32i)3-s + (0.890 − 0.454i)5-s + (1.85 + 0.497i)7-s + (−0.557 + 1.53i)9-s + (0.515 − 0.893i)11-s + (0.484 + 0.339i)13-s + (−1.43 − 0.760i)15-s + (−0.639 + 1.37i)17-s + (0.357 − 0.933i)19-s + (−1.06 − 2.93i)21-s + (−0.469 − 0.0410i)23-s + (0.586 − 0.809i)25-s + (0.988 − 0.264i)27-s + (−1.02 − 0.372i)29-s + (−0.165 + 0.0957i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.194 + 0.980i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.194 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08233 - 0.888382i\)
\(L(\frac12)\) \(\approx\) \(1.08233 - 0.888382i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.99 + 1.01i)T \)
19 \( 1 + (-1.56 + 4.07i)T \)
good3 \( 1 + (1.61 + 2.30i)T + (-1.02 + 2.81i)T^{2} \)
7 \( 1 + (-4.91 - 1.31i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.71 + 2.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.74 - 1.22i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (2.63 - 5.65i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (2.25 + 0.197i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (5.51 + 2.00i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.923 - 0.533i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.20 + 2.20i)T + 37iT^{2} \)
41 \( 1 + (2.29 + 0.404i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.102 + 1.17i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (2.83 - 1.32i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (1.09 - 12.4i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (-1.32 + 0.482i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (8.84 - 7.42i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.33 - 5.00i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (2.27 - 2.71i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.61 - 1.12i)T + (24.9 - 68.5i)T^{2} \)
79 \( 1 + (-1.00 + 5.67i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.0567 + 0.211i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (2.79 + 15.8i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-12.8 - 5.98i)T + (62.3 + 74.3i)T^{2} \)
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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

−11.30684190052030110060855289460, −10.72219351877027617710403506247, −8.942302078765003108604940268356, −8.409683222776430971739970749357, −7.32421380863137602909535209071, −6.07343204156737040679670978988, −5.69580056031578567849161015227, −4.52540927958524446860646058294, −2.01858132607102243640373431834, −1.31080954430567040851372631399, 1.74829908101440523454669368256, 3.79579162723462613558455182793, 4.89234553316751114955933024607, 5.35739414339995210991742015052, 6.63778728134678133916649502514, 7.79965503674060278982171848652, 9.168241230959931124657998728507, 9.923378636335361496645706595121, 10.69859973157312195977871401372, 11.30318811183478855093292360438

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