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Properties

Label 2-520-104.101-c1-0-10
Degree $2$
Conductor $520$
Sign $0.917 - 0.397i$
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  + (−1.24 − 0.669i)2-s + (−2.59 + 1.49i)3-s + (1.10 + 1.66i)4-s − 5-s + (4.23 − 0.128i)6-s + (−0.668 − 0.385i)7-s + (−0.257 − 2.81i)8-s + (2.98 − 5.16i)9-s + (1.24 + 0.669i)10-s + (0.183 + 0.317i)11-s + (−5.35 − 2.67i)12-s + (−2.71 − 2.36i)13-s + (0.574 + 0.928i)14-s + (2.59 − 1.49i)15-s + (−1.56 + 3.68i)16-s + (1.37 − 2.37i)17-s + ⋯
L(s)  = 1  + (−0.880 − 0.473i)2-s + (−1.49 + 0.864i)3-s + (0.551 + 0.834i)4-s − 0.447·5-s + (1.72 − 0.0525i)6-s + (−0.252 − 0.145i)7-s + (−0.0911 − 0.995i)8-s + (0.994 − 1.72i)9-s + (0.393 + 0.211i)10-s + (0.0552 + 0.0956i)11-s + (−1.54 − 0.771i)12-s + (−0.753 − 0.657i)13-s + (0.153 + 0.248i)14-s + (0.669 − 0.386i)15-s + (−0.391 + 0.920i)16-s + (0.332 − 0.575i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.399823 + 0.0828691i\)
\(L(\frac12)\) \(\approx\) \(0.399823 + 0.0828691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.669i)T \)
5 \( 1 + T \)
13 \( 1 + (2.71 + 2.36i)T \)
good3 \( 1 + (2.59 - 1.49i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.668 + 0.385i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.183 - 0.317i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.37 + 2.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.199 - 0.345i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.390 + 0.676i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.82 + 1.05i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.83iT - 31T^{2} \)
37 \( 1 + (-4.55 - 7.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.57 + 3.79i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.38 - 3.10i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.57iT - 47T^{2} \)
53 \( 1 + 4.85iT - 53T^{2} \)
59 \( 1 + (-6.27 + 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.23 - 0.712i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.10 - 3.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.22 - 0.707i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 16.3iT - 73T^{2} \)
79 \( 1 + 4.89T + 79T^{2} \)
83 \( 1 - 2.70T + 83T^{2} \)
89 \( 1 + (10.0 - 5.82i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.2 - 6.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.92965539200928051049301081009, −10.02015324749927022610244460372, −9.687749270966646275447590009993, −8.391046963813284686333732205388, −7.27803871036506343188789810422, −6.44159044929553179640209603234, −5.22339292356978175674787377440, −4.24187551814913355895296024280, −3.02832494781368753241627728914, −0.75287887081118375401535926109, 0.64134454576838714749813726514, 2.15262117236275886992444410208, 4.48862053096954855986474601595, 5.71999483110753094163265802037, 6.23982683455060525040749905597, 7.28978094072647894073211456724, 7.71352832253405875316080711042, 9.024609824169323005593419231669, 10.02051018443913870652362499786, 10.93231433691617764792891963728

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