Properties

Label 2-380-20.19-c2-0-42
Degree $2$
Conductor $380$
Sign $0.810 - 0.585i$
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  + (1.86 − 0.734i)2-s − 2.97·3-s + (2.92 − 2.73i)4-s + (−0.632 + 4.95i)5-s + (−5.52 + 2.18i)6-s + 5.57·7-s + (3.42 − 7.23i)8-s − 0.174·9-s + (2.46 + 9.69i)10-s + 6.76i·11-s + (−8.67 + 8.12i)12-s + 13.4i·13-s + (10.3 − 4.10i)14-s + (1.87 − 14.7i)15-s + (1.05 − 15.9i)16-s + 15.9i·17-s + ⋯
L(s)  = 1  + (0.930 − 0.367i)2-s − 0.990·3-s + (0.730 − 0.683i)4-s + (−0.126 + 0.991i)5-s + (−0.920 + 0.363i)6-s + 0.797·7-s + (0.427 − 0.903i)8-s − 0.0194·9-s + (0.246 + 0.969i)10-s + 0.614i·11-s + (−0.722 + 0.676i)12-s + 1.03i·13-s + (0.741 − 0.292i)14-s + (0.125 − 0.982i)15-s + (0.0658 − 0.997i)16-s + 0.935i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.810 - 0.585i$
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: \((2,\ 380,\ (\ :1),\ 0.810 - 0.585i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.02925 + 0.656397i\)
\(L(\frac12)\) \(\approx\) \(2.02925 + 0.656397i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.86 + 0.734i)T \)
5 \( 1 + (0.632 - 4.95i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 + 2.97T + 9T^{2} \)
7 \( 1 - 5.57T + 49T^{2} \)
11 \( 1 - 6.76iT - 121T^{2} \)
13 \( 1 - 13.4iT - 169T^{2} \)
17 \( 1 - 15.9iT - 289T^{2} \)
23 \( 1 - 37.1T + 529T^{2} \)
29 \( 1 - 45.3T + 841T^{2} \)
31 \( 1 - 10.3iT - 961T^{2} \)
37 \( 1 - 73.5iT - 1.36e3T^{2} \)
41 \( 1 + 73.2T + 1.68e3T^{2} \)
43 \( 1 - 69.3T + 1.84e3T^{2} \)
47 \( 1 + 41.1T + 2.20e3T^{2} \)
53 \( 1 + 75.3iT - 2.80e3T^{2} \)
59 \( 1 + 64.6iT - 3.48e3T^{2} \)
61 \( 1 - 52.6T + 3.72e3T^{2} \)
67 \( 1 + 40.0T + 4.48e3T^{2} \)
71 \( 1 + 2.17iT - 5.04e3T^{2} \)
73 \( 1 - 81.2iT - 5.32e3T^{2} \)
79 \( 1 + 89.5iT - 6.24e3T^{2} \)
83 \( 1 + 105.T + 6.88e3T^{2} \)
89 \( 1 + 9.02T + 7.92e3T^{2} \)
97 \( 1 - 36.2iT - 9.40e3T^{2} \)
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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

−11.46113130700514500618589073910, −10.66635525298066400563648233467, −9.964170133395322109698804193182, −8.314592503309806525002669738571, −6.82537214062524258663912645349, −6.52139702932731936722597099490, −5.21097367342682507152513099322, −4.45893581910802492518121566508, −3.05167495635631265242746673359, −1.60108062105680290379549836852, 0.853084453222873548627019666361, 2.92186627106333570790412499774, 4.53086732646606743610044781338, 5.22378779615020068214824779414, 5.80574040236670199445954153908, 7.09471373313604746060292981581, 8.127850677597469895439430390863, 8.930766525712527241538971011493, 10.64966704993873448585540372148, 11.29180339964437406977196868197

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