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Properties

Label 2-380-19.17-c1-0-5
Degree $2$
Conductor $380$
Sign $0.438 + 0.898i$
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  + (0.137 + 0.115i)3-s + (0.939 − 0.342i)5-s + (−1.55 − 2.69i)7-s + (−0.515 − 2.92i)9-s + (0.132 − 0.229i)11-s + (1.46 − 1.22i)13-s + (0.168 + 0.0612i)15-s + (0.886 − 5.02i)17-s + (0.524 + 4.32i)19-s + (0.0967 − 0.548i)21-s + (2.59 + 0.944i)23-s + (0.766 − 0.642i)25-s + (0.534 − 0.925i)27-s + (0.236 + 1.34i)29-s + (−0.337 − 0.585i)31-s + ⋯
L(s)  = 1  + (0.0791 + 0.0664i)3-s + (0.420 − 0.152i)5-s + (−0.588 − 1.01i)7-s + (−0.171 − 0.974i)9-s + (0.0399 − 0.0692i)11-s + (0.405 − 0.340i)13-s + (0.0434 + 0.0158i)15-s + (0.214 − 1.21i)17-s + (0.120 + 0.992i)19-s + (0.0211 − 0.119i)21-s + (0.540 + 0.196i)23-s + (0.153 − 0.128i)25-s + (0.102 − 0.178i)27-s + (0.0440 + 0.249i)29-s + (−0.0606 − 0.105i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.438 + 0.898i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.438 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14059 - 0.712624i\)
\(L(\frac12)\) \(\approx\) \(1.14059 - 0.712624i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.524 - 4.32i)T \)
good3 \( 1 + (-0.137 - 0.115i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (1.55 + 2.69i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.132 + 0.229i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.46 + 1.22i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.886 + 5.02i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-2.59 - 0.944i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.236 - 1.34i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.337 + 0.585i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.05T + 37T^{2} \)
41 \( 1 + (-1.29 - 1.08i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (6.23 - 2.27i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.49 + 8.45i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (2.22 + 0.809i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (2.24 - 12.7i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-3.92 - 1.42i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.447 - 2.53i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (6.34 - 2.30i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-8.15 - 6.84i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-8.28 - 6.95i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-1.66 - 2.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (12.5 - 10.4i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-2.07 + 11.7i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.17096206462150627198738725771, −10.05047757498900157847803704830, −9.578766908829544538354420734404, −8.507928646503999516455750544565, −7.30879374269766443488966116681, −6.47020860450611696914202045590, −5.42589463808551985275603985450, −4.00033935175952605188941851930, −3.04657664805646813412278680229, −0.950288671172903691803572162156, 2.00097613360553872101340253586, 3.11711329114302707110600659694, 4.73545711941434488171775002432, 5.82622414231187411230775344031, 6.60348252736710221023046457993, 7.908359452567111475799955771469, 8.823009364209456445851517726139, 9.596424025105975555260244423343, 10.67354136109322371200140160184, 11.39764635847283184308310142140

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