Properties

Label 2-380-380.59-c1-0-3
Degree $2$
Conductor $380$
Sign $-0.998 - 0.0528i$
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  + (−1.41 − 0.0466i)2-s + (−1.17 + 1.39i)3-s + (1.99 + 0.131i)4-s + (−1.08 − 1.95i)5-s + (1.72 − 1.92i)6-s + (1.36 + 2.36i)7-s + (−2.81 − 0.279i)8-s + (−0.0573 − 0.325i)9-s + (1.44 + 2.81i)10-s + (−2.24 − 1.29i)11-s + (−2.52 + 2.63i)12-s + (−0.527 + 0.443i)13-s + (−1.82 − 3.41i)14-s + (4.00 + 0.777i)15-s + (3.96 + 0.526i)16-s + (0.544 + 0.0960i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0329i)2-s + (−0.677 + 0.807i)3-s + (0.997 + 0.0659i)4-s + (−0.484 − 0.874i)5-s + (0.703 − 0.784i)6-s + (0.517 + 0.895i)7-s + (−0.995 − 0.0987i)8-s + (−0.0191 − 0.108i)9-s + (0.455 + 0.890i)10-s + (−0.675 − 0.389i)11-s + (−0.728 + 0.760i)12-s + (−0.146 + 0.122i)13-s + (−0.487 − 0.912i)14-s + (1.03 + 0.200i)15-s + (0.991 + 0.131i)16-s + (0.132 + 0.0232i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.998 - 0.0528i$
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: \((2,\ 380,\ (\ :1/2),\ -0.998 - 0.0528i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00530705 + 0.200581i\)
\(L(\frac12)\) \(\approx\) \(0.00530705 + 0.200581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0466i)T \)
5 \( 1 + (1.08 + 1.95i)T \)
19 \( 1 + (1.26 - 4.16i)T \)
good3 \( 1 + (1.17 - 1.39i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-1.36 - 2.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.24 + 1.29i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.527 - 0.443i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.544 - 0.0960i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.366 - 0.133i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (8.63 - 1.52i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (4.91 + 8.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.48T + 37T^{2} \)
41 \( 1 + (3.19 - 3.81i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.46 - 0.898i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.347 + 1.97i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-9.32 - 3.39i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.22 - 6.94i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (1.23 + 0.449i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-3.08 + 0.543i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (3.62 - 1.31i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (5.31 - 6.33i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-9.44 - 7.92i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.643 + 1.11i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.29 - 8.69i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.36 + 7.75i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.56611159683383098688677524550, −10.87985736144813931560579835420, −9.965320872568169162650912669671, −9.054959288759025160110999977835, −8.277604134590174648445497772850, −7.48685577791518097883905164928, −5.73625733947728850719018309440, −5.29040787537540002755395336823, −3.81973380420924143691226602681, −1.95829574573404776883481915982, 0.19018798271175371194260117962, 1.88756118670957902704294614280, 3.49329734927512076826447711636, 5.30340484586303639515081802942, 6.65324361651699261859821238819, 7.21809169915301678666672594682, 7.73655831455356387838899043036, 9.004322755830811342258070255386, 10.35123119022200955567263279615, 10.79578013904853396496273253031

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