Properties

Label 2-380-20.19-c2-0-38
Degree $2$
Conductor $380$
Sign $0.967 - 0.253i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.75 − 0.968i)2-s − 4.24·3-s + (2.12 + 3.38i)4-s + (4.77 + 1.49i)5-s + (7.43 + 4.11i)6-s + 7.86·7-s + (−0.437 − 7.98i)8-s + 9.03·9-s + (−6.90 − 7.23i)10-s + 2.63i·11-s + (−9.02 − 14.3i)12-s − 5.11i·13-s + (−13.7 − 7.61i)14-s + (−20.2 − 6.34i)15-s + (−6.96 + 14.4i)16-s + 12.2i·17-s + ⋯
L(s)  = 1  + (−0.875 − 0.484i)2-s − 1.41·3-s + (0.531 + 0.847i)4-s + (0.954 + 0.298i)5-s + (1.23 + 0.685i)6-s + 1.12·7-s + (−0.0547 − 0.998i)8-s + 1.00·9-s + (−0.690 − 0.723i)10-s + 0.239i·11-s + (−0.752 − 1.19i)12-s − 0.393i·13-s + (−0.982 − 0.543i)14-s + (−1.35 − 0.423i)15-s + (−0.435 + 0.900i)16-s + 0.721i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.967 - 0.253i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.967 - 0.253i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.870643 + 0.112345i\)
\(L(\frac12)\) \(\approx\) \(0.870643 + 0.112345i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.75 + 0.968i)T \)
5 \( 1 + (-4.77 - 1.49i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 + 4.24T + 9T^{2} \)
7 \( 1 - 7.86T + 49T^{2} \)
11 \( 1 - 2.63iT - 121T^{2} \)
13 \( 1 + 5.11iT - 169T^{2} \)
17 \( 1 - 12.2iT - 289T^{2} \)
23 \( 1 + 17.2T + 529T^{2} \)
29 \( 1 - 4.95T + 841T^{2} \)
31 \( 1 + 59.4iT - 961T^{2} \)
37 \( 1 - 47.8iT - 1.36e3T^{2} \)
41 \( 1 + 4.38T + 1.68e3T^{2} \)
43 \( 1 - 51.2T + 1.84e3T^{2} \)
47 \( 1 - 41.3T + 2.20e3T^{2} \)
53 \( 1 - 42.9iT - 2.80e3T^{2} \)
59 \( 1 - 103. iT - 3.48e3T^{2} \)
61 \( 1 - 68.7T + 3.72e3T^{2} \)
67 \( 1 - 69.7T + 4.48e3T^{2} \)
71 \( 1 - 55.1iT - 5.04e3T^{2} \)
73 \( 1 + 45.6iT - 5.32e3T^{2} \)
79 \( 1 - 84.1iT - 6.24e3T^{2} \)
83 \( 1 + 2.16T + 6.88e3T^{2} \)
89 \( 1 - 89.6T + 7.92e3T^{2} \)
97 \( 1 - 36.5iT - 9.40e3T^{2} \)
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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

−11.04584967697274513227765356869, −10.42051287075740503846245065255, −9.714579580898157388997945694422, −8.443455610110484352603838671425, −7.47544968354664802801318171096, −6.30951796941446258911904567284, −5.58401263989777181636939462185, −4.27265961567366514001032175674, −2.30869652094321298720347715394, −1.09683578176366578012072231457, 0.78270469412292431550184586853, 2.01740571673977854146858624277, 4.84254499452783362037748758663, 5.38958462749984113291004729663, 6.29405182858351476901588063967, 7.16914246801844064520918191490, 8.402134153750271365665454852042, 9.284507262752573641306792908937, 10.31062269681363199521035949600, 10.97313488305284323052525732458

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