Properties

Label 2-380-380.219-c1-0-21
Degree $2$
Conductor $380$
Sign $0.530 + 0.847i$
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  + (−0.660 − 1.25i)2-s + (0.742 + 0.884i)3-s + (−1.12 + 1.65i)4-s + (−2.17 + 0.500i)5-s + (0.615 − 1.51i)6-s + (1.69 − 2.94i)7-s + (2.81 + 0.316i)8-s + (0.289 − 1.64i)9-s + (2.06 + 2.39i)10-s + (−1.00 + 0.581i)11-s + (−2.29 + 0.230i)12-s + (4.39 + 3.68i)13-s + (−4.79 − 0.179i)14-s + (−2.05 − 1.55i)15-s + (−1.46 − 3.72i)16-s + (0.599 − 0.105i)17-s + ⋯
L(s)  = 1  + (−0.467 − 0.884i)2-s + (0.428 + 0.510i)3-s + (−0.563 + 0.826i)4-s + (−0.974 + 0.223i)5-s + (0.251 − 0.617i)6-s + (0.641 − 1.11i)7-s + (0.993 + 0.111i)8-s + (0.0965 − 0.547i)9-s + (0.653 + 0.757i)10-s + (−0.303 + 0.175i)11-s + (−0.663 + 0.0664i)12-s + (1.21 + 1.02i)13-s + (−1.28 − 0.0479i)14-s + (−0.531 − 0.401i)15-s + (−0.365 − 0.930i)16-s + (0.145 − 0.0256i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.976735 - 0.541260i\)
\(L(\frac12)\) \(\approx\) \(0.976735 - 0.541260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.660 + 1.25i)T \)
5 \( 1 + (2.17 - 0.500i)T \)
19 \( 1 + (-3.23 + 2.91i)T \)
good3 \( 1 + (-0.742 - 0.884i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (-1.69 + 2.94i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.00 - 0.581i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.39 - 3.68i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.599 + 0.105i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-7.77 + 2.82i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (6.74 + 1.18i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.13 + 3.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.94T + 37T^{2} \)
41 \( 1 + (-5.38 - 6.41i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.97 + 1.08i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.606 + 3.43i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (8.80 - 3.20i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.65 - 9.41i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (1.14 - 0.415i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-6.92 - 1.22i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (14.5 + 5.29i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.55 - 1.85i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (1.09 - 0.919i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.12 - 5.41i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.76 + 4.48i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.0622 - 0.353i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.18779339387284699337503056392, −10.47593886935456911713977163816, −9.353931518762222744938552267915, −8.668502712797096190474564456311, −7.66386126769109438355536937026, −6.88923933621768955957428179653, −4.62471698599379587653535926318, −3.99628430740327331531867317341, −3.08514557947913627777229580554, −1.05885130969764405911543977320, 1.40608109968470755849974328609, 3.31122739065111207195694147868, 5.03434583106528579163157760936, 5.64209586796882760816885348503, 7.14710822761076098943235972792, 7.946297390663619273318932371653, 8.409933506360436646133586627194, 9.172454366779351723197745413572, 10.68330810753069865997125657382, 11.30490967041885378513054511030

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