Properties

Label 2-380-20.19-c2-0-85
Degree $2$
Conductor $380$
Sign $0.0891 + 0.996i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.68 − 1.07i)2-s + 5.94·3-s + (1.70 + 3.61i)4-s + (1.71 − 4.69i)5-s + (−10.0 − 6.36i)6-s − 9.59·7-s + (0.995 − 7.93i)8-s + 26.3·9-s + (−7.93 + 6.08i)10-s − 5.60i·11-s + (10.1 + 21.5i)12-s + 1.46i·13-s + (16.2 + 10.2i)14-s + (10.2 − 27.9i)15-s + (−10.1 + 12.3i)16-s − 30.4i·17-s + ⋯
L(s)  = 1  + (−0.844 − 0.535i)2-s + 1.98·3-s + (0.426 + 0.904i)4-s + (0.343 − 0.938i)5-s + (−1.67 − 1.06i)6-s − 1.37·7-s + (0.124 − 0.992i)8-s + 2.92·9-s + (−0.793 + 0.608i)10-s − 0.509i·11-s + (0.845 + 1.79i)12-s + 0.112i·13-s + (1.15 + 0.734i)14-s + (0.681 − 1.86i)15-s + (−0.636 + 0.771i)16-s − 1.79i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0891 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0891 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0891 + 0.996i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.0891 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.56222 - 1.42862i\)
\(L(\frac12)\) \(\approx\) \(1.56222 - 1.42862i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.68 + 1.07i)T \)
5 \( 1 + (-1.71 + 4.69i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 5.94T + 9T^{2} \)
7 \( 1 + 9.59T + 49T^{2} \)
11 \( 1 + 5.60iT - 121T^{2} \)
13 \( 1 - 1.46iT - 169T^{2} \)
17 \( 1 + 30.4iT - 289T^{2} \)
23 \( 1 - 5.38T + 529T^{2} \)
29 \( 1 - 25.0T + 841T^{2} \)
31 \( 1 - 26.0iT - 961T^{2} \)
37 \( 1 - 35.8iT - 1.36e3T^{2} \)
41 \( 1 - 3.18T + 1.68e3T^{2} \)
43 \( 1 - 57.1T + 1.84e3T^{2} \)
47 \( 1 + 30.6T + 2.20e3T^{2} \)
53 \( 1 - 61.6iT - 2.80e3T^{2} \)
59 \( 1 - 50.5iT - 3.48e3T^{2} \)
61 \( 1 + 31.5T + 3.72e3T^{2} \)
67 \( 1 + 13.4T + 4.48e3T^{2} \)
71 \( 1 + 23.8iT - 5.04e3T^{2} \)
73 \( 1 + 63.7iT - 5.32e3T^{2} \)
79 \( 1 - 57.1iT - 6.24e3T^{2} \)
83 \( 1 + 71.3T + 6.88e3T^{2} \)
89 \( 1 + 7.54T + 7.92e3T^{2} \)
97 \( 1 - 118. iT - 9.40e3T^{2} \)
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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

−10.39456259241335350972272354305, −9.549873828097327095157206253651, −9.202304641201662133016313361667, −8.536425913071082510313935688434, −7.53602807152907767626482019827, −6.64223599143447395558452517073, −4.49079569835569182416783337311, −3.21273558134585234264134817519, −2.61804312666577944544468292101, −1.03886585826179640204788807983, 1.88836723401864686621581810764, 2.87405688700080045599657758176, 3.94597499463280069343538949477, 6.14321168050505389880844447215, 6.91258457597811721123685923723, 7.72531099620410435744210427654, 8.600437290955119757422839473657, 9.545608385147051555718840054336, 9.973669600094329927265551498945, 10.68939597361837890601237807502

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