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Properties

Label 2-380-76.75-c1-0-16
Degree $2$
Conductor $380$
Sign $0.999 + 0.0401i$
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  + (0.611 + 1.27i)2-s − 3.21·3-s + (−1.25 + 1.56i)4-s − 5-s + (−1.96 − 4.09i)6-s − 3.02i·7-s + (−2.75 − 0.640i)8-s + 7.30·9-s + (−0.611 − 1.27i)10-s + 0.488i·11-s + (4.01 − 5.00i)12-s + 0.943i·13-s + (3.86 − 1.85i)14-s + 3.21·15-s + (−0.869 − 3.90i)16-s + 3.16·17-s + ⋯
L(s)  = 1  + (0.432 + 0.901i)2-s − 1.85·3-s + (−0.625 + 0.780i)4-s − 0.447·5-s + (−0.802 − 1.67i)6-s − 1.14i·7-s + (−0.974 − 0.226i)8-s + 2.43·9-s + (−0.193 − 0.403i)10-s + 0.147i·11-s + (1.15 − 1.44i)12-s + 0.261i·13-s + (1.03 − 0.495i)14-s + 0.828·15-s + (−0.217 − 0.976i)16-s + 0.768·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.999 + 0.0401i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.999 + 0.0401i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656889 - 0.0132056i\)
\(L(\frac12)\) \(\approx\) \(0.656889 - 0.0132056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.611 - 1.27i)T \)
5 \( 1 + T \)
19 \( 1 + (-2.58 - 3.50i)T \)
good3 \( 1 + 3.21T + 3T^{2} \)
7 \( 1 + 3.02iT - 7T^{2} \)
11 \( 1 - 0.488iT - 11T^{2} \)
13 \( 1 - 0.943iT - 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
23 \( 1 + 7.08iT - 23T^{2} \)
29 \( 1 + 8.80iT - 29T^{2} \)
31 \( 1 - 3.66T + 31T^{2} \)
37 \( 1 + 1.11iT - 37T^{2} \)
41 \( 1 + 1.56iT - 41T^{2} \)
43 \( 1 + 3.58iT - 43T^{2} \)
47 \( 1 + 0.471iT - 47T^{2} \)
53 \( 1 + 12.4iT - 53T^{2} \)
59 \( 1 - 8.65T + 59T^{2} \)
61 \( 1 + 8.62T + 61T^{2} \)
67 \( 1 - 6.59T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 - 18.0iT - 83T^{2} \)
89 \( 1 + 9.83iT - 89T^{2} \)
97 \( 1 - 1.71iT - 97T^{2} \)
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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

−11.63243835859235152439799397376, −10.47790436352777607000948153749, −9.831322871897721076693818353974, −8.093197660574282481931757920803, −7.24294958325015541798759374234, −6.52319899726024356034810913544, −5.62361747231039967292867636966, −4.58824984405315109016891189364, −3.90074567189436327756897746572, −0.58431504779852795483341951671, 1.22680949381164664525122907570, 3.18856635045298862397709835385, 4.68811938656326369100345557851, 5.42513602520180865371535220466, 6.06411948018139883449444481572, 7.36590472200099869722020569281, 8.963324313718629535603607613886, 9.913521663534725923864234837187, 10.82137067560280870879643736464, 11.54850749016161661356154092848

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