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Properties

Label 2-520-13.3-c1-0-1
Degree $2$
Conductor $520$
Sign $0.477 - 0.878i$
Analytic cond. $4.15222$
Root an. cond. $2.03769$
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 − 5-s + (−1.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (2.5 + 4.33i)11-s + (−3.5 − 0.866i)13-s + (1 + 1.73i)15-s + (1 − 1.73i)17-s + (−2.5 + 4.33i)19-s + 6·21-s + (4 + 6.92i)23-s + 25-s − 4.00·27-s + 8·31-s + (5 − 8.66i)33-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s − 0.447·5-s + (−0.566 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.753 + 1.30i)11-s + (−0.970 − 0.240i)13-s + (0.258 + 0.447i)15-s + (0.242 − 0.420i)17-s + (−0.573 + 0.993i)19-s + 1.30·21-s + (0.834 + 1.44i)23-s + 0.200·25-s − 0.769·27-s + 1.43·31-s + (0.870 − 1.50i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637050 + 0.378775i\)
\(L(\frac12)\) \(\approx\) \(0.637050 + 0.378775i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + (3.5 + 0.866i)T \)
good3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 + 13T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7 - 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 18T + 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.48008873255910635651723886650, −9.907816353503547255066002309107, −9.454624674402232144645692317687, −8.122855747745404338458844020915, −7.24544842307836986278041309078, −6.57095188727646719945827884715, −5.62506602365759226129265588190, −4.46494788291630552346851699379, −2.92457759081347430146599083645, −1.52910091486766373339193764318, 0.48916504237269301481936092421, 3.02698115803248446284436623256, 4.17731667075657223208493401789, 4.73497446674590201702950542505, 6.14249378485309992354479616544, 6.90939505675490837371576179864, 8.095143182554253306387423651825, 9.164250711330399335538261173856, 9.946769463376995820593405009809, 10.91561360834698325611942938550

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