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Properties

Label 2-380-19.7-c1-0-5
Degree $2$
Conductor $380$
Sign $0.360 + 0.932i$
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  + (−0.354 − 0.614i)3-s + (−0.5 − 0.866i)5-s + 3.11·7-s + (1.24 − 2.16i)9-s − 3.52·11-s + (−0.200 + 0.347i)13-s + (−0.354 + 0.614i)15-s + (−1.74 − 3.02i)17-s + (4.35 − 0.251i)19-s + (−1.10 − 1.91i)21-s + (3.65 − 6.33i)23-s + (−0.499 + 0.866i)25-s − 3.89·27-s + (3.96 − 6.86i)29-s + 5.73·31-s + ⋯
L(s)  = 1  + (−0.204 − 0.354i)3-s + (−0.223 − 0.387i)5-s + 1.17·7-s + (0.416 − 0.720i)9-s − 1.06·11-s + (−0.0556 + 0.0964i)13-s + (−0.0915 + 0.158i)15-s + (−0.424 − 0.734i)17-s + (0.998 − 0.0577i)19-s + (−0.240 − 0.416i)21-s + (0.762 − 1.32i)23-s + (−0.0999 + 0.173i)25-s − 0.750·27-s + (0.736 − 1.27i)29-s + 1.03·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.360 + 0.932i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.360 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07532 - 0.737442i\)
\(L(\frac12)\) \(\approx\) \(1.07532 - 0.737442i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-4.35 + 0.251i)T \)
good3 \( 1 + (0.354 + 0.614i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 3.11T + 7T^{2} \)
11 \( 1 + 3.52T + 11T^{2} \)
13 \( 1 + (0.200 - 0.347i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.74 + 3.02i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.65 + 6.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.96 + 6.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.73T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 + (-0.555 - 0.961i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.30 - 7.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.76 - 6.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.27 - 9.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.25 - 7.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.61 + 7.98i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.20 - 7.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.31 - 7.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.870 + 1.50i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.50 - 7.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.07T + 83T^{2} \)
89 \( 1 + (-6.61 + 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.85 - 6.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.36130138634138808513985031500, −10.35339171649876155706031931121, −9.291921410225189693146220660729, −8.250419959199405651928533687454, −7.53800808556912802475421025199, −6.47727010442478896766648228469, −5.13996140523067507593566820827, −4.44402740886320771401966039019, −2.68729767403263981844063335742, −0.990413314656452807125653544747, 1.84213141740888498894496777583, 3.41937975074693589557824707093, 4.89827486425458345349048421789, 5.32531640845324913121052485521, 7.01216042936862670405579627020, 7.80636194875614772006068489193, 8.594068927210827818089321642360, 10.00015561887179724818012301001, 10.69957960919140263787281294738, 11.28028638652935778823611564418

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