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Properties

Label 2-380-76.31-c1-0-21
Degree $2$
Conductor $380$
Sign $0.200 - 0.979i$
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  + (1.35 + 0.394i)2-s + (1.50 + 2.59i)3-s + (1.68 + 1.07i)4-s + (−0.5 − 0.866i)5-s + (1.01 + 4.12i)6-s − 4.06i·7-s + (1.87 + 2.12i)8-s + (−3.00 + 5.20i)9-s + (−0.337 − 1.37i)10-s + 3.65i·11-s + (−0.249 + 5.99i)12-s + (−1.47 − 0.854i)13-s + (1.60 − 5.51i)14-s + (1.50 − 2.59i)15-s + (1.70 + 3.61i)16-s + (−3.34 − 5.79i)17-s + ⋯
L(s)  = 1  + (0.960 + 0.278i)2-s + (0.866 + 1.50i)3-s + (0.844 + 0.535i)4-s + (−0.223 − 0.387i)5-s + (0.413 + 1.68i)6-s − 1.53i·7-s + (0.661 + 0.749i)8-s + (−1.00 + 1.73i)9-s + (−0.106 − 0.434i)10-s + 1.10i·11-s + (−0.0720 + 1.73i)12-s + (−0.410 − 0.236i)13-s + (0.428 − 1.47i)14-s + (0.387 − 0.671i)15-s + (0.426 + 0.904i)16-s + (−0.810 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.200 - 0.979i$
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: \((2,\ 380,\ (\ :1/2),\ 0.200 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22601 + 1.81728i\)
\(L(\frac12)\) \(\approx\) \(2.22601 + 1.81728i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.394i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.270 + 4.35i)T \)
good3 \( 1 + (-1.50 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 4.06iT - 7T^{2} \)
11 \( 1 - 3.65iT - 11T^{2} \)
13 \( 1 + (1.47 + 0.854i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.34 + 5.79i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.43 - 1.40i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.738 + 0.426i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.54T + 31T^{2} \)
37 \( 1 + 2.39iT - 37T^{2} \)
41 \( 1 + (7.85 - 4.53i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.53 + 2.61i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-11.5 - 6.64i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.43 - 1.40i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.0742 + 0.128i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.45 + 4.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.62 - 2.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.02 + 10.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.443 + 0.767i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.61 - 6.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 + (4.29 + 2.47i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.44 - 3.72i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.35701704818512924089633285205, −10.69319987500101769650909718446, −9.752299049750184557099131830785, −8.928483527892354849000293695318, −7.55441965825841060645298192436, −7.09154880223661041389225744181, −5.05468302547986537974849086681, −4.56508005017904532611840521684, −3.80282064406386178736742815484, −2.61913334203946926473798624243, 1.81670713895637342013142721113, 2.65948698893551490246370045172, 3.68485184764239806123083925656, 5.64085071179702242636784918235, 6.26831347647477783718432729052, 7.22156209290182661134390436254, 8.346451774504602816939398315969, 8.949524864549776816163638204218, 10.55815950273292308616315927028, 11.65801779138008749807918038735

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