Properties

Label 2-380-380.59-c1-0-0
Degree $2$
Conductor $380$
Sign $-0.862 - 0.505i$
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  + (−1.24 − 0.670i)2-s + (0.497 − 0.592i)3-s + (1.10 + 1.66i)4-s + (−1.92 + 1.14i)5-s + (−1.01 + 0.404i)6-s + (−0.647 − 1.12i)7-s + (−0.252 − 2.81i)8-s + (0.417 + 2.36i)9-s + (3.15 − 0.132i)10-s + (−3.51 − 2.03i)11-s + (1.53 + 0.177i)12-s + (0.500 − 0.419i)13-s + (0.0545 + 1.82i)14-s + (−0.279 + 1.70i)15-s + (−1.57 + 3.67i)16-s + (−6.09 − 1.07i)17-s + ⋯
L(s)  = 1  + (−0.880 − 0.473i)2-s + (0.286 − 0.342i)3-s + (0.550 + 0.834i)4-s + (−0.859 + 0.510i)5-s + (−0.414 + 0.165i)6-s + (−0.244 − 0.423i)7-s + (−0.0893 − 0.995i)8-s + (0.139 + 0.788i)9-s + (0.999 − 0.0417i)10-s + (−1.06 − 0.612i)11-s + (0.443 + 0.0511i)12-s + (0.138 − 0.116i)13-s + (0.0145 + 0.488i)14-s + (−0.0722 + 0.440i)15-s + (−0.393 + 0.919i)16-s + (−1.47 − 0.260i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.862 - 0.505i$
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: \((2,\ 380,\ (\ :1/2),\ -0.862 - 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00179522 + 0.00661855i\)
\(L(\frac12)\) \(\approx\) \(0.00179522 + 0.00661855i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.670i)T \)
5 \( 1 + (1.92 - 1.14i)T \)
19 \( 1 + (4.32 - 0.521i)T \)
good3 \( 1 + (-0.497 + 0.592i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.647 + 1.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.51 + 2.03i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.500 + 0.419i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (6.09 + 1.07i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (3.27 + 1.19i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.235 + 0.0414i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.17 - 3.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + (3.07 - 3.66i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.67 - 0.608i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-2.14 - 12.1i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (1.04 + 0.381i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.57 + 8.93i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-7.73 - 2.81i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (8.22 - 1.45i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.30 - 0.837i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-7.87 + 9.38i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-10.3 - 8.72i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (5.50 + 9.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.55 + 11.3i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.139 + 0.790i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.41084616880535040527418187114, −10.68980313719070571875787334887, −10.27003778131071907764354213867, −8.655626568418492682243239766479, −8.198691652170875036088963827321, −7.29468289876906519617680495128, −6.50690367081283388568299916938, −4.55557687064006454117391759525, −3.26670287466164288557018877509, −2.20692820663505471792546168105, 0.00539556242248712742408419930, 2.25144939314031425237520361847, 3.97232518878377042831827446149, 5.12770642807778440575860158677, 6.44050535892060929515594745217, 7.32695689091125915590047490560, 8.540095623506203658033855031749, 8.802834324046669936628420529830, 9.932843219438739644011072157071, 10.72001984668252490140161489571

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