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Properties

Label 2-520-104.69-c1-0-11
Degree $2$
Conductor $520$
Sign $0.984 + 0.176i$
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  + (−0.417 − 1.35i)2-s + (−1.00 − 0.580i)3-s + (−1.65 + 1.12i)4-s + 5-s + (−0.364 + 1.60i)6-s + (−2.74 + 1.58i)7-s + (2.21 + 1.76i)8-s + (−0.826 − 1.43i)9-s + (−0.417 − 1.35i)10-s + (−1.23 + 2.13i)11-s + (2.31 − 0.175i)12-s + (3.60 + 0.150i)13-s + (3.28 + 3.04i)14-s + (−1.00 − 0.580i)15-s + (1.45 − 3.72i)16-s + (0.369 + 0.639i)17-s + ⋯
L(s)  = 1  + (−0.295 − 0.955i)2-s + (−0.580 − 0.335i)3-s + (−0.825 + 0.564i)4-s + 0.447·5-s + (−0.148 + 0.653i)6-s + (−1.03 + 0.598i)7-s + (0.782 + 0.622i)8-s + (−0.275 − 0.477i)9-s + (−0.132 − 0.427i)10-s + (−0.371 + 0.643i)11-s + (0.668 − 0.0506i)12-s + (0.999 + 0.0416i)13-s + (0.877 + 0.813i)14-s + (−0.259 − 0.149i)15-s + (0.363 − 0.931i)16-s + (0.0895 + 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.984 + 0.176i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.984 + 0.176i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.789850 - 0.0700978i\)
\(L(\frac12)\) \(\approx\) \(0.789850 - 0.0700978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.417 + 1.35i)T \)
5 \( 1 - T \)
13 \( 1 + (-3.60 - 0.150i)T \)
good3 \( 1 + (1.00 + 0.580i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.74 - 1.58i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.23 - 2.13i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.369 - 0.639i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.31 - 7.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.88 + 4.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.59 - 0.919i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.05iT - 31T^{2} \)
37 \( 1 + (-1.35 + 2.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.165 - 0.0952i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.43 + 4.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.23iT - 47T^{2} \)
53 \( 1 - 4.18iT - 53T^{2} \)
59 \( 1 + (-5.72 - 9.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.03 - 2.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.66 - 2.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.24 - 1.87i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.03iT - 73T^{2} \)
79 \( 1 + 9.18T + 79T^{2} \)
83 \( 1 - 1.79T + 83T^{2} \)
89 \( 1 + (-12.1 - 7.04i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.8 - 8.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.70408593149410961255691857440, −10.14719640343093552784391773900, −9.186102311886725946143713394255, −8.554145540660360025307055018713, −7.18072753894081425954899925087, −6.10278547030343444839680252753, −5.33789166170732443574379479574, −3.75501491186510882235311734467, −2.77182955663918343285431579113, −1.23923588230200569902590736599, 0.65891489725841402378564813176, 3.13805770120695634449426061513, 4.54511139913823067298989013877, 5.57855847370836146455397900055, 6.18277077057870354647374849099, 7.16777087203406723294038441384, 8.117923761009004618101914412745, 9.294603444756853887049708910927, 9.763780151243386260499590309574, 10.86436116292131413821118348168

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