Properties

Label 2-380-380.59-c1-0-52
Degree $2$
Conductor $380$
Sign $-0.977 - 0.211i$
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  + (−0.690 − 1.23i)2-s + (2.07 − 2.47i)3-s + (−1.04 + 1.70i)4-s + (−2.16 − 0.568i)5-s + (−4.48 − 0.853i)6-s + (−1.40 − 2.44i)7-s + (2.82 + 0.113i)8-s + (−1.28 − 7.31i)9-s + (0.791 + 3.06i)10-s + (3.57 + 2.06i)11-s + (2.04 + 6.12i)12-s + (−0.432 + 0.363i)13-s + (−2.03 + 3.42i)14-s + (−5.89 + 4.16i)15-s + (−1.81 − 3.56i)16-s + (−6.16 − 1.08i)17-s + ⋯
L(s)  = 1  + (−0.488 − 0.872i)2-s + (1.19 − 1.42i)3-s + (−0.523 + 0.852i)4-s + (−0.967 − 0.254i)5-s + (−1.83 − 0.348i)6-s + (−0.532 − 0.922i)7-s + (0.999 + 0.0402i)8-s + (−0.429 − 2.43i)9-s + (0.250 + 0.968i)10-s + (1.07 + 0.622i)11-s + (0.590 + 1.76i)12-s + (−0.120 + 0.100i)13-s + (−0.544 + 0.915i)14-s + (−1.52 + 1.07i)15-s + (−0.452 − 0.891i)16-s + (−1.49 − 0.263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.977 - 0.211i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116584 + 1.09216i\)
\(L(\frac12)\) \(\approx\) \(0.116584 + 1.09216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.690 + 1.23i)T \)
5 \( 1 + (2.16 + 0.568i)T \)
19 \( 1 + (-3.10 - 3.06i)T \)
good3 \( 1 + (-2.07 + 2.47i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (1.40 + 2.44i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.57 - 2.06i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.432 - 0.363i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (6.16 + 1.08i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.24 - 0.451i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (2.13 - 0.376i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.48 + 4.30i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.45T + 37T^{2} \)
41 \( 1 + (-2.98 + 3.55i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-7.73 + 2.81i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.670 + 3.80i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (7.29 + 2.65i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.57 + 8.92i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-7.95 - 2.89i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-4.11 + 0.725i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.36 - 1.58i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-0.244 + 0.291i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-6.60 - 5.54i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.842 + 1.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.63 - 6.71i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.0553 - 0.313i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.12647441219845201677788212272, −9.584641911581104160979894250360, −9.061450423655140760482086727400, −8.077193520036143027760884035365, −7.30278877034685328987388510852, −6.80170433235948214075649764497, −4.15170495895178751178350180711, −3.51846996888275082319163365738, −2.11877820731923578068644714221, −0.78098008143989527472466556194, 2.78614067747703337776656714313, 3.94923659715978804420123762273, 4.81285453964253973155845640521, 6.19824611277096688196810004746, 7.41875272171606668616148123436, 8.475827595755244200195156814889, 9.056192370744110591683178137298, 9.467910764370308411172003080635, 10.76131438546602429517773266317, 11.36929395627153382906660474887

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