Properties

Label 2-380-380.59-c1-0-20
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: $\chi_{380} (59, \cdot )$
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

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

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