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Properties

Label 2-380-20.7-c1-0-41
Degree $2$
Conductor $380$
Sign $-0.749 + 0.662i$
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  + (−1.34 + 0.433i)2-s + (−0.497 − 0.497i)3-s + (1.62 − 1.16i)4-s + (−2.01 − 0.978i)5-s + (0.885 + 0.454i)6-s + (2.91 − 2.91i)7-s + (−1.68 + 2.27i)8-s − 2.50i·9-s + (3.13 + 0.445i)10-s + 3.58i·11-s + (−1.38 − 0.227i)12-s + (−3.29 + 3.29i)13-s + (−2.66 + 5.19i)14-s + (0.514 + 1.48i)15-s + (1.27 − 3.79i)16-s + (−4.60 − 4.60i)17-s + ⋯
L(s)  = 1  + (−0.951 + 0.306i)2-s + (−0.287 − 0.287i)3-s + (0.812 − 0.583i)4-s + (−0.899 − 0.437i)5-s + (0.361 + 0.185i)6-s + (1.10 − 1.10i)7-s + (−0.594 + 0.804i)8-s − 0.834i·9-s + (0.990 + 0.140i)10-s + 1.08i·11-s + (−0.401 − 0.0658i)12-s + (−0.914 + 0.914i)13-s + (−0.712 + 1.38i)14-s + (0.132 + 0.384i)15-s + (0.319 − 0.947i)16-s + (−1.11 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149871 - 0.395984i\)
\(L(\frac12)\) \(\approx\) \(0.149871 - 0.395984i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.433i)T \)
5 \( 1 + (2.01 + 0.978i)T \)
19 \( 1 - T \)
good3 \( 1 + (0.497 + 0.497i)T + 3iT^{2} \)
7 \( 1 + (-2.91 + 2.91i)T - 7iT^{2} \)
11 \( 1 - 3.58iT - 11T^{2} \)
13 \( 1 + (3.29 - 3.29i)T - 13iT^{2} \)
17 \( 1 + (4.60 + 4.60i)T + 17iT^{2} \)
23 \( 1 + (3.51 + 3.51i)T + 23iT^{2} \)
29 \( 1 + 4.88iT - 29T^{2} \)
31 \( 1 - 1.57iT - 31T^{2} \)
37 \( 1 + (5.48 + 5.48i)T + 37iT^{2} \)
41 \( 1 + 3.69T + 41T^{2} \)
43 \( 1 + (4.56 + 4.56i)T + 43iT^{2} \)
47 \( 1 + (1.60 - 1.60i)T - 47iT^{2} \)
53 \( 1 + (-2.12 + 2.12i)T - 53iT^{2} \)
59 \( 1 - 5.91T + 59T^{2} \)
61 \( 1 - 0.536T + 61T^{2} \)
67 \( 1 + (1.64 - 1.64i)T - 67iT^{2} \)
71 \( 1 + 4.08iT - 71T^{2} \)
73 \( 1 + (0.480 - 0.480i)T - 73iT^{2} \)
79 \( 1 + 3.35T + 79T^{2} \)
83 \( 1 + (-12.4 - 12.4i)T + 83iT^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 + (6.21 + 6.21i)T + 97iT^{2} \)
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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

−11.08247758131505934892229848783, −9.974668448702255984391249708874, −9.111449817691872381929888787487, −8.109649731089409969721973338717, −7.07697568364563169112322615390, −6.95608184081477950278016194722, −5.01647748607684047461924256662, −4.19392577566444478867505032661, −1.93842696766361451218818755240, −0.38318552093554829784538544050, 2.05502346619493860053889334171, 3.31550664679321495821900597179, 4.86836192643384069281895341148, 5.99740091944476417034267820027, 7.40855275779258793042071048661, 8.287784068924730305613673444596, 8.588898672681881567355918907331, 10.13620722673060134720055090366, 10.85424031516393357389714497541, 11.48635718971469426924579630664

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