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Properties

Label 2-520-520.229-c0-0-0
Degree $2$
Conductor $520$
Sign $-0.471 - 0.881i$
Analytic cond. $0.259513$
Root an. cond. $0.509424$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s − 1.41·3-s + 1.00i·4-s + (0.707 + 0.707i)5-s + (−1.00 − 1.00i)6-s + (−0.707 + 0.707i)8-s + 1.00·9-s + 1.00i·10-s − 1.41i·12-s + (−0.707 + 0.707i)13-s + (−1.00 − 1.00i)15-s − 1.00·16-s + (0.707 + 0.707i)18-s + (−0.707 + 0.707i)20-s + (1.00 − 1.00i)24-s + 1.00i·25-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s − 1.41·3-s + 1.00i·4-s + (0.707 + 0.707i)5-s + (−1.00 − 1.00i)6-s + (−0.707 + 0.707i)8-s + 1.00·9-s + 1.00i·10-s − 1.41i·12-s + (−0.707 + 0.707i)13-s + (−1.00 − 1.00i)15-s − 1.00·16-s + (0.707 + 0.707i)18-s + (−0.707 + 0.707i)20-s + (1.00 − 1.00i)24-s + 1.00i·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.471 - 0.881i$
Analytic conductor: \(0.259513\)
Root analytic conductor: \(0.509424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :0),\ -0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8482869160\)
\(L(\frac12)\) \(\approx\) \(0.8482869160\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + 1.41T + T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1 - i)T + iT^{2} \)
37 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
41 \( 1 + (1 + i)T + iT^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
71 \( 1 + (-1 - i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
89 \( 1 + (1 - i)T - iT^{2} \)
97 \( 1 - iT^{2} \)
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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

−11.56475956445263178538248366526, −10.68274601739905940283696439035, −9.801892861575323300244659003203, −8.604366487208339400691388281602, −7.07656708765407702565093523761, −6.80416035290831779692484178303, −5.73140692045159020498111508002, −5.20254679182702523009154648941, −4.01346263281715644405446921792, −2.43122042023343294751633204992, 1.06454893369197427891116770885, 2.68947656260577915368783303652, 4.48489976789066035358130920611, 5.06889828262744358691758693796, 5.93567182759199885174685633127, 6.54837456300973257899977906103, 8.143550739591182183939445460222, 9.657429593536567794530622792878, 9.996020146486643528939220760451, 11.02592584326493617322183052208

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