Properties

Label 2-1520-19.7-c1-0-17
Degree $2$
Conductor $1520$
Sign $0.0977 - 0.995i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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 + 1.73i)3-s + (0.5 + 0.866i)5-s + 4·7-s + (−0.499 + 0.866i)9-s + 3·11-s + (−3 + 5.19i)13-s + (−0.999 + 1.73i)15-s + (−1 − 1.73i)17-s + (−3.5 + 2.59i)19-s + (4 + 6.92i)21-s + (2 − 3.46i)23-s + (−0.499 + 0.866i)25-s + 4.00·27-s + (−0.5 + 0.866i)29-s + 5·31-s + ⋯
L(s)  = 1  + (0.577 + 0.999i)3-s + (0.223 + 0.387i)5-s + 1.51·7-s + (−0.166 + 0.288i)9-s + 0.904·11-s + (−0.832 + 1.44i)13-s + (−0.258 + 0.447i)15-s + (−0.242 − 0.420i)17-s + (−0.802 + 0.596i)19-s + (0.872 + 1.51i)21-s + (0.417 − 0.722i)23-s + (−0.0999 + 0.173i)25-s + 0.769·27-s + (−0.0928 + 0.160i)29-s + 0.898·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.0977 - 0.995i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.606915778\)
\(L(\frac12)\) \(\approx\) \(2.606915778\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 2.59i)T \)
good3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.5 - 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6 + 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (8.5 - 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6 + 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.665108650282069722653549891607, −8.778270963584984394363042247224, −8.448103209241532077401077917019, −7.16357483589614806200098256720, −6.58914289596137312124726065867, −5.23147247484025996508583531477, −4.41385406118851039570089918388, −3.98875876297654614203503397045, −2.58281633659935237285442425199, −1.63856205374735701460078038991, 1.07006181674947654735909188159, 1.90481701583711419969009404120, 2.87151794214658261779283704543, 4.35740711008743360675183931725, 5.06529070346488421490569925502, 6.05698972656116033746548650923, 7.14004226091595057598861284147, 7.72460010183171537284386502854, 8.411201090297281374490953571761, 8.921335094335021783608039767149

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