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Properties

Label 2-380-95.33-c1-0-0
Degree $2$
Conductor $380$
Sign $-0.582 - 0.812i$
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  + (0.935 + 1.33i)3-s + (−0.351 + 2.20i)5-s + (−4.10 − 1.10i)7-s + (0.116 − 0.320i)9-s + (−3.07 + 5.33i)11-s + (4.00 + 2.80i)13-s + (−3.27 + 1.59i)15-s + (−1.67 + 3.58i)17-s + (3.36 − 2.77i)19-s + (−2.37 − 6.51i)21-s + (−3.80 − 0.332i)23-s + (−4.75 − 1.55i)25-s + (5.26 − 1.41i)27-s + (−2.08 − 0.759i)29-s + (2.56 − 1.47i)31-s + ⋯
L(s)  = 1  + (0.539 + 0.771i)3-s + (−0.157 + 0.987i)5-s + (−1.55 − 0.416i)7-s + (0.0389 − 0.106i)9-s + (−0.928 + 1.60i)11-s + (1.11 + 0.778i)13-s + (−0.846 + 0.412i)15-s + (−0.405 + 0.868i)17-s + (0.771 − 0.635i)19-s + (−0.517 − 1.42i)21-s + (−0.792 − 0.0693i)23-s + (−0.950 − 0.310i)25-s + (1.01 − 0.271i)27-s + (−0.387 − 0.141i)29-s + (0.460 − 0.265i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.582 - 0.812i$
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: \((2,\ 380,\ (\ :1/2),\ -0.582 - 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.527500 + 1.02709i\)
\(L(\frac12)\) \(\approx\) \(0.527500 + 1.02709i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.351 - 2.20i)T \)
19 \( 1 + (-3.36 + 2.77i)T \)
good3 \( 1 + (-0.935 - 1.33i)T + (-1.02 + 2.81i)T^{2} \)
7 \( 1 + (4.10 + 1.10i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.07 - 5.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.00 - 2.80i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (1.67 - 3.58i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (3.80 + 0.332i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (2.08 + 0.759i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.56 + 1.47i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.93 - 2.93i)T + 37iT^{2} \)
41 \( 1 + (-11.8 - 2.08i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.317 - 3.62i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (5.68 - 2.64i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-0.309 + 3.53i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (-10.3 + 3.75i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.42 + 2.03i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.868 - 1.86i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (1.80 - 2.15i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.0922 + 0.0645i)T + (24.9 - 68.5i)T^{2} \)
79 \( 1 + (-1.95 + 11.0i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.25 - 8.40i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.149 + 0.849i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-11.9 - 5.59i)T + (62.3 + 74.3i)T^{2} \)
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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

−11.49091685243439486153638105111, −10.44217112722606667840881892774, −9.848078070331789382203912863820, −9.319425975944496020636599871155, −7.901440345391076981915383493731, −6.84648495056879487574001807352, −6.21955007537392844715241765879, −4.34340129837451053091070970734, −3.62160436473914163308762497128, −2.53990157526067130025425043832, 0.72055925077902371902260407514, 2.67833571655673032069221326795, 3.62486518015402760425073820807, 5.48102057244690068680400565682, 6.07975014414394832130325861092, 7.49477028883331889582736713063, 8.313746427406535530715280738848, 8.942997371489819195942810692931, 9.988746576801517444360563034982, 11.09903737491076624201509212571

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