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Properties

Label 2-760-19.7-c1-0-15
Degree $2$
Conductor $760$
Sign $-0.345 + 0.938i$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
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  + (−0.832 − 1.44i)3-s + (−0.5 − 0.866i)5-s + 1.93·7-s + (0.114 − 0.198i)9-s + 1.38·11-s + (0.362 − 0.627i)13-s + (−0.832 + 1.44i)15-s + (1.38 + 2.39i)17-s + (0.607 − 4.31i)19-s + (−1.60 − 2.78i)21-s + (4.34 − 7.52i)23-s + (−0.499 + 0.866i)25-s − 5.37·27-s + (−2.76 + 4.79i)29-s − 2.37·31-s + ⋯
L(s)  = 1  + (−0.480 − 0.832i)3-s + (−0.223 − 0.387i)5-s + 0.729·7-s + (0.0382 − 0.0661i)9-s + 0.418·11-s + (0.100 − 0.174i)13-s + (−0.214 + 0.372i)15-s + (0.335 + 0.581i)17-s + (0.139 − 0.990i)19-s + (−0.350 − 0.607i)21-s + (0.905 − 1.56i)23-s + (−0.0999 + 0.173i)25-s − 1.03·27-s + (−0.513 + 0.890i)29-s − 0.426·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.345 + 0.938i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ -0.345 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.737938 - 1.05801i\)
\(L(\frac12)\) \(\approx\) \(0.737938 - 1.05801i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.607 + 4.31i)T \)
good3 \( 1 + (0.832 + 1.44i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 1.93T + 7T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 + (-0.362 + 0.627i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.38 - 2.39i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.34 + 7.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.76 - 4.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.37T + 31T^{2} \)
37 \( 1 + 6.58T + 37T^{2} \)
41 \( 1 + (-1.62 - 2.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.67 + 8.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.78 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.64 + 6.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.56 - 2.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.44 - 2.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.435 + 0.753i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.16 - 2.02i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.02 + 3.50i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.16 - 7.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.97T + 83T^{2} \)
89 \( 1 + (2.41 - 4.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.77 - 6.53i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.22789466451498202597432562618, −8.931407926588416368077076562319, −8.452215844702225522410216112316, −7.24756638951690800750170347932, −6.78020752548045789912756375206, −5.61398795629522000637125072687, −4.76998730190785858400660269330, −3.57779237357833421933888189701, −1.92152161919928797601117453211, −0.75379024652668632679098345037, 1.63606301629739767067938242280, 3.32680509967531593206291900077, 4.26490867757936288665456250046, 5.17885461428416380772579849218, 5.96382433129294015784403372263, 7.30119254425693140532929474147, 7.86677983349995246989216002504, 9.127008608447449694308610977153, 9.812556933839912378058765783891, 10.66719942447569694291967828372

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