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Properties

Label 2-380-76.75-c1-0-15
Degree $2$
Conductor $380$
Sign $0.452 + 0.891i$
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  + (−1.31 + 0.521i)2-s − 2.15·3-s + (1.45 − 1.37i)4-s − 5-s + (2.83 − 1.12i)6-s + 1.66i·7-s + (−1.19 + 2.56i)8-s + 1.65·9-s + (1.31 − 0.521i)10-s + 1.77i·11-s + (−3.14 + 2.96i)12-s − 0.769i·13-s + (−0.866 − 2.18i)14-s + 2.15·15-s + (0.237 − 3.99i)16-s − 3.23·17-s + ⋯
L(s)  = 1  + (−0.929 + 0.368i)2-s − 1.24·3-s + (0.727 − 0.685i)4-s − 0.447·5-s + (1.15 − 0.459i)6-s + 0.628i·7-s + (−0.423 + 0.905i)8-s + 0.553·9-s + (0.415 − 0.164i)10-s + 0.535i·11-s + (−0.907 + 0.854i)12-s − 0.213i·13-s + (−0.231 − 0.583i)14-s + 0.557·15-s + (0.0594 − 0.998i)16-s − 0.785·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.452 + 0.891i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.452 + 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.277618 - 0.170532i\)
\(L(\frac12)\) \(\approx\) \(0.277618 - 0.170532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.521i)T \)
5 \( 1 + T \)
19 \( 1 + (-1.23 + 4.18i)T \)
good3 \( 1 + 2.15T + 3T^{2} \)
7 \( 1 - 1.66iT - 7T^{2} \)
11 \( 1 - 1.77iT - 11T^{2} \)
13 \( 1 + 0.769iT - 13T^{2} \)
17 \( 1 + 3.23T + 17T^{2} \)
23 \( 1 + 3.06iT - 23T^{2} \)
29 \( 1 + 6.45iT - 29T^{2} \)
31 \( 1 - 4.09T + 31T^{2} \)
37 \( 1 + 5.34iT - 37T^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 + 11.5iT - 43T^{2} \)
47 \( 1 - 6.81iT - 47T^{2} \)
53 \( 1 + 5.59iT - 53T^{2} \)
59 \( 1 + 5.35T + 59T^{2} \)
61 \( 1 + 4.94T + 61T^{2} \)
67 \( 1 + 6.72T + 67T^{2} \)
71 \( 1 - 1.83T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 9.41T + 79T^{2} \)
83 \( 1 + 7.48iT - 83T^{2} \)
89 \( 1 - 16.0iT - 89T^{2} \)
97 \( 1 - 0.676iT - 97T^{2} \)
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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

−11.10292725709096867699278150115, −10.42991586469361882040890224240, −9.318355160736912707770848050625, −8.475553971860451607811404167084, −7.32172048132926396645505670940, −6.48872749371179508367549227124, −5.60695829640293113298138592558, −4.61664585289902820057778396575, −2.42496577051881569543558957632, −0.39426465030013406628130772413, 1.18228384049603217173853112163, 3.24940924734220094005799669009, 4.56346652384265720111748155518, 6.00252957791142251738795044804, 6.83518639466272848764854267244, 7.81201949240053052848403781798, 8.783371846712105022825543202590, 9.961306405516913252622643576580, 10.73951723769140032514929122548, 11.36775919107406635939839834057

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