Properties

Label 2-380-19.6-c1-0-3
Degree $2$
Conductor $380$
Sign $0.915 - 0.403i$
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  + (2.40 + 0.874i)3-s + (−0.173 − 0.984i)5-s + (−0.118 + 0.204i)7-s + (2.70 + 2.27i)9-s + (2.60 + 4.51i)11-s + (−1.93 + 0.705i)13-s + (0.443 − 2.51i)15-s + (4.62 − 3.88i)17-s + (1.86 − 3.93i)19-s + (−0.462 + 0.387i)21-s + (−0.685 + 3.88i)23-s + (−0.939 + 0.342i)25-s + (0.682 + 1.18i)27-s + (−6.90 − 5.79i)29-s + (−1.86 + 3.23i)31-s + ⋯
L(s)  = 1  + (1.38 + 0.504i)3-s + (−0.0776 − 0.440i)5-s + (−0.0446 + 0.0772i)7-s + (0.902 + 0.757i)9-s + (0.786 + 1.36i)11-s + (−0.537 + 0.195i)13-s + (0.114 − 0.649i)15-s + (1.12 − 0.941i)17-s + (0.427 − 0.903i)19-s + (−0.100 + 0.0846i)21-s + (−0.142 + 0.810i)23-s + (−0.187 + 0.0684i)25-s + (0.131 + 0.227i)27-s + (−1.28 − 1.07i)29-s + (−0.335 + 0.580i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.915 - 0.403i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.915 - 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03678 + 0.429106i\)
\(L(\frac12)\) \(\approx\) \(2.03678 + 0.429106i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-1.86 + 3.93i)T \)
good3 \( 1 + (-2.40 - 0.874i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (0.118 - 0.204i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.60 - 4.51i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.93 - 0.705i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-4.62 + 3.88i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.685 - 3.88i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (6.90 + 5.79i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.86 - 3.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.73T + 37T^{2} \)
41 \( 1 + (11.1 + 4.04i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.0112 + 0.0636i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (6.65 + 5.58i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.300 + 1.70i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-0.134 + 0.112i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.78 - 10.0i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.759 + 0.637i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.860 + 4.87i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (2.27 + 0.829i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-6.14 - 2.23i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.39 + 4.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.51 - 2.00i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (2.70 - 2.27i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.60611501026152485328740637723, −9.946773398334591196070954045382, −9.554952156266562332739116448528, −8.892974315276730678010750391603, −7.70236761089811863287981064126, −7.09269930170388708600225877419, −5.28604206070965539478834698093, −4.28895901768994878773425827976, −3.23915074122860386116354838456, −1.91992793169750117884057449540, 1.62823592693087152323691328506, 3.15846528580881537394034895266, 3.70582510743853166119183029380, 5.64554975885369966276154009065, 6.72714208649912467859708807898, 7.82397725323890559996828969308, 8.333868222961957038262467793780, 9.315278610398933476805718164859, 10.23420533653025992283196682530, 11.31339959835176038863763420014

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