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Properties

Label 2-520-104.101-c1-0-20
Degree $2$
Conductor $520$
Sign $0.995 - 0.0981i$
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  + (−1.28 − 0.584i)2-s + (1.00 − 0.582i)3-s + (1.31 + 1.50i)4-s + 5-s + (−1.63 + 0.160i)6-s + (0.786 + 0.454i)7-s + (−0.814 − 2.70i)8-s + (−0.821 + 1.42i)9-s + (−1.28 − 0.584i)10-s + (2.01 + 3.49i)11-s + (2.20 + 0.752i)12-s + (1.21 + 3.39i)13-s + (−0.747 − 1.04i)14-s + (1.00 − 0.582i)15-s + (−0.534 + 3.96i)16-s + (1.41 − 2.45i)17-s + ⋯
L(s)  = 1  + (−0.910 − 0.413i)2-s + (0.582 − 0.336i)3-s + (0.658 + 0.752i)4-s + 0.447·5-s + (−0.669 + 0.0653i)6-s + (0.297 + 0.171i)7-s + (−0.287 − 0.957i)8-s + (−0.273 + 0.474i)9-s + (−0.407 − 0.184i)10-s + (0.608 + 1.05i)11-s + (0.636 + 0.217i)12-s + (0.336 + 0.941i)13-s + (−0.199 − 0.279i)14-s + (0.260 − 0.150i)15-s + (−0.133 + 0.991i)16-s + (0.344 − 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.995 - 0.0981i$
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: \((2,\ 520,\ (\ :1/2),\ 0.995 - 0.0981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28832 + 0.0634096i\)
\(L(\frac12)\) \(\approx\) \(1.28832 + 0.0634096i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 0.584i)T \)
5 \( 1 - T \)
13 \( 1 + (-1.21 - 3.39i)T \)
good3 \( 1 + (-1.00 + 0.582i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.786 - 0.454i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.01 - 3.49i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.41 + 2.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.810 + 1.40i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.494 + 0.856i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.87 - 4.54i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.84iT - 31T^{2} \)
37 \( 1 + (0.538 + 0.931i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.68 + 4.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.23 - 5.33i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.5iT - 47T^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 + (-4.67 + 8.09i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.159 - 0.0918i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.74 - 9.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.27 - 2.46i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.44iT - 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + (2.39 - 1.38i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.80 + 5.66i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.89736484737994082725460456664, −9.736187258768394320499776935261, −9.163798247384157760122386708096, −8.418741942237630982741676771178, −7.34492425798007092348027958920, −6.80006702695614874053175734675, −5.27743834900158006038107525059, −3.81882981946944870138486567506, −2.40870401918101502524462616558, −1.63386182501573039001668036273, 1.07080453804646296672384968134, 2.73414942019696714429865181044, 3.94386493210342530711859813451, 5.84958487006861920083479725292, 6.01055565669897889898866472337, 7.61513557412249130985893025874, 8.226342651561169086295388740634, 9.165276378782644931231078736891, 9.642998337597152318722140954064, 10.74858291997980612318396566449

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