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Properties

Label 2-520-65.33-c1-0-0
Degree $2$
Conductor $520$
Sign $-0.804 + 0.593i$
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  + (−2.82 + 0.756i)3-s + (1.25 + 1.84i)5-s + (−2.78 + 1.60i)7-s + (4.79 − 2.77i)9-s + (0.777 + 2.89i)11-s + (−3.60 + 0.150i)13-s + (−4.95 − 4.26i)15-s + (0.786 − 2.93i)17-s + (2.14 + 0.575i)19-s + (6.65 − 6.65i)21-s + (−0.454 − 1.69i)23-s + (−1.82 + 4.65i)25-s + (−5.25 + 5.25i)27-s + (−3.28 − 1.89i)29-s + (−6.69 − 6.69i)31-s + ⋯
L(s)  = 1  + (−1.62 + 0.436i)3-s + (0.563 + 0.826i)5-s + (−1.05 + 0.608i)7-s + (1.59 − 0.923i)9-s + (0.234 + 0.874i)11-s + (−0.999 + 0.0418i)13-s + (−1.27 − 1.10i)15-s + (0.190 − 0.711i)17-s + (0.492 + 0.132i)19-s + (1.45 − 1.45i)21-s + (−0.0947 − 0.353i)23-s + (−0.365 + 0.930i)25-s + (−1.01 + 1.01i)27-s + (−0.610 − 0.352i)29-s + (−1.20 − 1.20i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.804 + 0.593i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.804 + 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0528398 - 0.160539i\)
\(L(\frac12)\) \(\approx\) \(0.0528398 - 0.160539i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 1.84i)T \)
13 \( 1 + (3.60 - 0.150i)T \)
good3 \( 1 + (2.82 - 0.756i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.78 - 1.60i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.777 - 2.89i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.786 + 2.93i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.14 - 0.575i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.454 + 1.69i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.28 + 1.89i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.69 + 6.69i)T + 31iT^{2} \)
37 \( 1 + (8.59 + 4.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.35 - 1.97i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.218 - 0.0584i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 9.57iT - 47T^{2} \)
53 \( 1 + (-9.06 - 9.06i)T + 53iT^{2} \)
59 \( 1 + (-1.26 + 4.70i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.55 - 9.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.660 + 1.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.14 + 4.26i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 7.68T + 73T^{2} \)
79 \( 1 - 1.08iT - 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (15.8 - 4.25i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.730 - 1.26i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.43498691749828286357322973673, −10.41952929669385215738095466320, −9.809065659885255388499442427682, −9.332879163923861019091193814773, −7.22844598547569036550734734304, −6.79363387451056986968364930681, −5.73002387665177131955771446682, −5.21852912602598626897622417785, −3.80552709867597851790327749948, −2.31035180870513116576184466231, 0.12261943926252176434116337534, 1.46423490364281461224806345476, 3.58665877507938517261495533484, 5.05565512924583940057017226949, 5.60902063120487002580446249371, 6.56915378247798785586605306770, 7.22226375852163122061463191448, 8.624453926817090107279802573861, 9.768720219642199160190302836586, 10.34094302114888379385428541516

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