Properties

Label 2-380-380.239-c2-0-26
Degree $2$
Conductor $380$
Sign $-0.664 - 0.747i$
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  + (1.38 + 1.43i)2-s + (0.480 − 0.831i)3-s + (−0.143 + 3.99i)4-s + (−1.72 − 4.69i)5-s + (1.86 − 0.463i)6-s − 5.58·7-s + (−5.95 + 5.34i)8-s + (4.03 + 6.99i)9-s + (4.35 − 9.00i)10-s + 16.9i·11-s + (3.25 + 2.03i)12-s + (−4.31 + 2.48i)13-s + (−7.74 − 8.03i)14-s + (−4.73 − 0.815i)15-s + (−15.9 − 1.15i)16-s + (20.3 + 11.7i)17-s + ⋯
L(s)  = 1  + (0.694 + 0.719i)2-s + (0.160 − 0.277i)3-s + (−0.0359 + 0.999i)4-s + (−0.345 − 0.938i)5-s + (0.310 − 0.0772i)6-s − 0.797·7-s + (−0.744 + 0.667i)8-s + (0.448 + 0.777i)9-s + (0.435 − 0.900i)10-s + 1.54i·11-s + (0.271 + 0.169i)12-s + (−0.331 + 0.191i)13-s + (−0.553 − 0.573i)14-s + (−0.315 − 0.0543i)15-s + (−0.997 − 0.0719i)16-s + (1.19 + 0.692i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.664 - 0.747i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ -0.664 - 0.747i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.720823 + 1.60449i\)
\(L(\frac12)\) \(\approx\) \(0.720823 + 1.60449i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 1.43i)T \)
5 \( 1 + (1.72 + 4.69i)T \)
19 \( 1 + (14.2 - 12.5i)T \)
good3 \( 1 + (-0.480 + 0.831i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + 5.58T + 49T^{2} \)
11 \( 1 - 16.9iT - 121T^{2} \)
13 \( 1 + (4.31 - 2.48i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-20.3 - 11.7i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (-1.57 - 2.73i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-15.8 - 27.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 7.75iT - 961T^{2} \)
37 \( 1 + 28.8iT - 1.36e3T^{2} \)
41 \( 1 + (-29.1 + 50.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-19.7 + 34.1i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (41.8 + 72.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (87.8 - 50.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-78.2 - 45.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (35.1 + 60.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-58.0 - 100. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (74.5 + 43.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-13.1 - 7.56i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-109. - 63.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 22.6T + 6.88e3T^{2} \)
89 \( 1 + (5.95 + 10.3i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (31.5 + 18.2i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.11769848966151517966711640705, −10.46815528927799134767046072406, −9.493663914563841120547279181863, −8.409838492148374916830428081514, −7.57677567158120271079240714197, −6.87456423877358061946653040998, −5.55320737318996026611645604923, −4.64550178239090588505379680730, −3.70046652116103389245565799077, −1.97749749473316179041571251410, 0.60425272309528801499451417498, 2.95842926703443045248751418344, 3.28963816942778801652778425952, 4.54867298427848985069329059341, 6.10527868972310896318662238886, 6.55751839047084392145245959162, 8.018359750071826445171149300262, 9.470437104795491049787262677604, 9.931869281251580494269623340935, 10.98279075939082167418424711097

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