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Properties

Label 2-380-76.31-c1-0-13
Degree $2$
Conductor $380$
Sign $0.925 - 0.377i$
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  + (−1.27 − 0.615i)2-s + (0.481 + 0.834i)3-s + (1.24 + 1.56i)4-s + (0.5 + 0.866i)5-s + (−0.100 − 1.35i)6-s + 1.00i·7-s + (−0.620 − 2.75i)8-s + (1.03 − 1.79i)9-s + (−0.104 − 1.41i)10-s − 4.53i·11-s + (−0.708 + 1.79i)12-s + (4.62 + 2.66i)13-s + (0.620 − 1.28i)14-s + (−0.481 + 0.834i)15-s + (−0.907 + 3.89i)16-s + (2.17 + 3.76i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.434i)2-s + (0.278 + 0.481i)3-s + (0.621 + 0.783i)4-s + (0.223 + 0.387i)5-s + (−0.0409 − 0.554i)6-s + 0.381i·7-s + (−0.219 − 0.975i)8-s + (0.345 − 0.597i)9-s + (−0.0329 − 0.446i)10-s − 1.36i·11-s + (−0.204 + 0.517i)12-s + (1.28 + 0.740i)13-s + (0.165 − 0.343i)14-s + (−0.124 + 0.215i)15-s + (−0.226 + 0.973i)16-s + (0.526 + 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.925 - 0.377i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.925 - 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08413 + 0.212519i\)
\(L(\frac12)\) \(\approx\) \(1.08413 + 0.212519i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 + 0.615i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (2.83 - 3.30i)T \)
good3 \( 1 + (-0.481 - 0.834i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 1.00iT - 7T^{2} \)
11 \( 1 + 4.53iT - 11T^{2} \)
13 \( 1 + (-4.62 - 2.66i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.17 - 3.76i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.38 - 1.37i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.30 - 1.91i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.27T + 31T^{2} \)
37 \( 1 - 3.89iT - 37T^{2} \)
41 \( 1 + (0.221 - 0.127i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.13 + 0.653i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.20 - 3.00i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.70 + 5.02i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.64 + 8.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.67 + 8.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.09 - 3.62i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.75 + 3.04i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.39 + 12.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.03 - 8.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.98iT - 83T^{2} \)
89 \( 1 + (0.0672 + 0.0388i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.0 - 8.08i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.02777063160885558638334232403, −10.59421567443462028516111029243, −9.485696695569604107114940458081, −8.787250399732420470743588772265, −8.122799240288706956226666702698, −6.63980020124189165555410880953, −5.96014173161030973468988307920, −3.86827588479947057645924375768, −3.22280717584414822436901367811, −1.48880344445143856698614699925, 1.17375932248336834006084881646, 2.48392976821933952649317332323, 4.55094143962072361570914882083, 5.67696453155681117067962697371, 6.97704819010046511639909654795, 7.49987693238235882017345305165, 8.490617586528094520875177632149, 9.316541701035573557738246396559, 10.33461970319151654145536933361, 10.92392850014167593380328867749

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