Properties

Label 2-380-76.31-c1-0-36
Degree $2$
Conductor $380$
Sign $-0.796 + 0.604i$
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  + (1.15 − 0.817i)2-s + (−1.05 − 1.82i)3-s + (0.664 − 1.88i)4-s + (0.5 + 0.866i)5-s + (−2.70 − 1.24i)6-s − 0.279i·7-s + (−0.773 − 2.72i)8-s + (−0.719 + 1.24i)9-s + (1.28 + 0.591i)10-s − 2.67i·11-s + (−4.14 + 0.773i)12-s + (−3.06 − 1.77i)13-s + (−0.227 − 0.322i)14-s + (1.05 − 1.82i)15-s + (−3.11 − 2.50i)16-s + (1.01 + 1.75i)17-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)2-s + (−0.608 − 1.05i)3-s + (0.332 − 0.943i)4-s + (0.223 + 0.387i)5-s + (−1.10 − 0.508i)6-s − 0.105i·7-s + (−0.273 − 0.961i)8-s + (−0.239 + 0.415i)9-s + (0.406 + 0.186i)10-s − 0.807i·11-s + (−1.19 + 0.223i)12-s + (−0.851 − 0.491i)13-s + (−0.0609 − 0.0860i)14-s + (0.271 − 0.471i)15-s + (−0.778 − 0.627i)16-s + (0.246 + 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.796 + 0.604i$
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: \((2,\ 380,\ (\ :1/2),\ -0.796 + 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540999 - 1.60748i\)
\(L(\frac12)\) \(\approx\) \(0.540999 - 1.60748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 0.817i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-2.32 - 3.68i)T \)
good3 \( 1 + (1.05 + 1.82i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 0.279iT - 7T^{2} \)
11 \( 1 + 2.67iT - 11T^{2} \)
13 \( 1 + (3.06 + 1.77i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.01 - 1.75i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.18 - 0.682i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.05 - 0.606i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.76T + 31T^{2} \)
37 \( 1 - 0.254iT - 37T^{2} \)
41 \( 1 + (-6.19 + 3.57i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.49 + 3.75i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.23 - 1.29i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.79 + 3.92i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.13 - 8.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.28 + 2.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.21 + 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.81 + 3.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.69 - 4.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.63 - 9.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.31iT - 83T^{2} \)
89 \( 1 + (-0.830 - 0.479i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.76 + 1.02i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.17759087821008036620483684478, −10.44217657569444503264845243673, −9.469393610163419622221388116530, −7.86530447423172995569135353088, −6.93723429985829776426805492046, −6.01288112505490263544864137931, −5.36316564040134104441485079681, −3.76016467451832279336975363920, −2.47627733662676041038057412271, −0.995067694858419260349361592082, 2.60202281693758223267733078173, 4.25082458696052049710491440850, 4.82023219470022456122117796346, 5.60401629264069454108548240873, 6.82918987200919876594346198376, 7.75641434134251669324017733242, 9.162932047927516077815637961536, 9.803726044680252926435755462208, 10.97440201369223556122975235288, 11.81151221108047063852369091258

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