Properties

Label 2-380-380.379-c1-0-1
Degree $2$
Conductor $380$
Sign $0.0971 + 0.995i$
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  + (−0.948 + 1.04i)2-s + 1.76i·3-s + (−0.201 − 1.98i)4-s + (−1.85 + 1.24i)5-s + (−1.84 − 1.67i)6-s − 4.29·7-s + (2.27 + 1.67i)8-s − 0.102·9-s + (0.457 − 3.12i)10-s + 0.376i·11-s + (3.50 − 0.355i)12-s + 0.928·13-s + (4.07 − 4.50i)14-s + (−2.19 − 3.27i)15-s + (−3.91 + 0.802i)16-s − 5.96i·17-s + ⋯
L(s)  = 1  + (−0.670 + 0.741i)2-s + 1.01i·3-s + (−0.100 − 0.994i)4-s + (−0.831 + 0.556i)5-s + (−0.754 − 0.681i)6-s − 1.62·7-s + (0.805 + 0.592i)8-s − 0.0343·9-s + (0.144 − 0.989i)10-s + 0.113i·11-s + (1.01 − 0.102i)12-s + 0.257·13-s + (1.08 − 1.20i)14-s + (−0.565 − 0.845i)15-s + (−0.979 + 0.200i)16-s − 1.44i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0971 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0971 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0971 + 0.995i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.0971 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0269478 - 0.0244451i\)
\(L(\frac12)\) \(\approx\) \(0.0269478 - 0.0244451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.948 - 1.04i)T \)
5 \( 1 + (1.85 - 1.24i)T \)
19 \( 1 + (3.61 + 2.43i)T \)
good3 \( 1 - 1.76iT - 3T^{2} \)
7 \( 1 + 4.29T + 7T^{2} \)
11 \( 1 - 0.376iT - 11T^{2} \)
13 \( 1 - 0.928T + 13T^{2} \)
17 \( 1 + 5.96iT - 17T^{2} \)
23 \( 1 - 0.352T + 23T^{2} \)
29 \( 1 - 3.89iT - 29T^{2} \)
31 \( 1 - 1.06T + 31T^{2} \)
37 \( 1 + 9.60T + 37T^{2} \)
41 \( 1 + 6.82iT - 41T^{2} \)
43 \( 1 + 9.45T + 43T^{2} \)
47 \( 1 + 2.56T + 47T^{2} \)
53 \( 1 - 3.18T + 53T^{2} \)
59 \( 1 - 3.98T + 59T^{2} \)
61 \( 1 + 3.22T + 61T^{2} \)
67 \( 1 - 8.75iT - 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 5.09iT - 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 3.49T + 83T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 + 9.74T + 97T^{2} \)
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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

−11.76980852513012085824222361950, −10.67626423804496387333698067032, −10.14933878013256244988037239488, −9.324768175354465243561008642427, −8.583869173701212872931203578032, −7.01520705755260887877581976907, −6.83996567564007587479674616444, −5.33621222954444767662839843030, −4.16449613253375894410807299206, −3.03036387332439882895092671846, 0.03017257209451202219036945415, 1.62142114614152698642667098961, 3.26257592970119739379784132735, 4.16437036121129078974589691740, 6.22399294427482276629859333584, 7.00026869747694042104945432285, 8.078041231883220914966798032559, 8.656307363523852146109646378221, 9.823318181774213704809541638850, 10.57629707382521951225035627637

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