Properties

Label 2-380-19.16-c1-0-1
Degree $2$
Conductor $380$
Sign $0.590 - 0.806i$
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  + (0.671 − 0.244i)3-s + (−0.173 + 0.984i)5-s + (1.15 + 2.00i)7-s + (−1.90 + 1.60i)9-s + (−1.57 + 2.72i)11-s + (5.20 + 1.89i)13-s + (0.124 + 0.703i)15-s + (−2.31 − 1.94i)17-s + (4.24 − 0.973i)19-s + (1.26 + 1.06i)21-s + (−0.283 − 1.60i)23-s + (−0.939 − 0.342i)25-s + (−1.96 + 3.39i)27-s + (3.91 − 3.28i)29-s + (1.01 + 1.75i)31-s + ⋯
L(s)  = 1  + (0.387 − 0.141i)3-s + (−0.0776 + 0.440i)5-s + (0.438 + 0.759i)7-s + (−0.635 + 0.533i)9-s + (−0.474 + 0.822i)11-s + (1.44 + 0.525i)13-s + (0.0320 + 0.181i)15-s + (−0.561 − 0.471i)17-s + (0.974 − 0.223i)19-s + (0.276 + 0.232i)21-s + (−0.0591 − 0.335i)23-s + (−0.187 − 0.0684i)25-s + (−0.377 + 0.653i)27-s + (0.726 − 0.609i)29-s + (0.181 + 0.314i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.590 - 0.806i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.590 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33533 + 0.677101i\)
\(L(\frac12)\) \(\approx\) \(1.33533 + 0.677101i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (-4.24 + 0.973i)T \)
good3 \( 1 + (-0.671 + 0.244i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.15 - 2.00i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.57 - 2.72i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.20 - 1.89i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.31 + 1.94i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.283 + 1.60i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.91 + 3.28i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.01 - 1.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.94T + 37T^{2} \)
41 \( 1 + (-6.75 + 2.45i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.0418 - 0.237i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (5.49 - 4.60i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (0.521 + 2.95i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (0.466 + 0.391i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.87 + 10.6i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (11.5 - 9.65i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.791 + 4.49i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-13.6 + 4.95i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (4.11 - 1.49i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (2.97 + 5.15i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.52 - 3.10i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-7.30 - 6.12i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.40332769511884625886332951033, −10.77750134569505701456422569955, −9.526679838614869705909684074688, −8.638401943163074509707089255804, −7.906896275033535263304376883376, −6.81555650393411654335773187657, −5.69217527976818043812004957282, −4.61457142181316106465380873164, −3.07517621199499142541106656353, −2.01164617801130439054672512185, 1.06175585440897607432692553074, 3.11927657673555234949755576098, 4.01680674861921510573139542152, 5.41265301213260728676062832727, 6.32424499192711113483015546248, 7.77929932164200700941459891032, 8.431984938338288085665980183629, 9.183401256495883831131360615655, 10.46398114994477316105659071996, 11.11280781362393132875018726138

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