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Properties

Label 2-507-13.12-c3-0-30
Degree $2$
Conductor $507$
Sign $0.246 - 0.969i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.86i·2-s − 3·3-s + 4.52·4-s + 2.36i·5-s − 5.59i·6-s − 4.86i·7-s + 23.3i·8-s + 9·9-s − 4.40·10-s − 35.1i·11-s − 13.5·12-s + 9.07·14-s − 7.08i·15-s − 7.35·16-s + 33.1·17-s + 16.7i·18-s + ⋯
L(s)  = 1  + 0.659i·2-s − 0.577·3-s + 0.565·4-s + 0.211i·5-s − 0.380i·6-s − 0.262i·7-s + 1.03i·8-s + 0.333·9-s − 0.139·10-s − 0.963i·11-s − 0.326·12-s + 0.173·14-s − 0.121i·15-s − 0.114·16-s + 0.472·17-s + 0.219i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ 0.246 - 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.981745180\)
\(L(\frac12)\) \(\approx\) \(1.981745180\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 \)
good2 \( 1 - 1.86iT - 8T^{2} \)
5 \( 1 - 2.36iT - 125T^{2} \)
7 \( 1 + 4.86iT - 343T^{2} \)
11 \( 1 + 35.1iT - 1.33e3T^{2} \)
17 \( 1 - 33.1T + 4.91e3T^{2} \)
19 \( 1 - 104. iT - 6.85e3T^{2} \)
23 \( 1 - 86.3T + 1.21e4T^{2} \)
29 \( 1 + 118.T + 2.43e4T^{2} \)
31 \( 1 + 262. iT - 2.97e4T^{2} \)
37 \( 1 + 59.6iT - 5.06e4T^{2} \)
41 \( 1 - 76.9iT - 6.89e4T^{2} \)
43 \( 1 - 344.T + 7.95e4T^{2} \)
47 \( 1 - 415. iT - 1.03e5T^{2} \)
53 \( 1 - 141.T + 1.48e5T^{2} \)
59 \( 1 - 598. iT - 2.05e5T^{2} \)
61 \( 1 - 791.T + 2.26e5T^{2} \)
67 \( 1 + 22.1iT - 3.00e5T^{2} \)
71 \( 1 - 599. iT - 3.57e5T^{2} \)
73 \( 1 - 776. iT - 3.89e5T^{2} \)
79 \( 1 + 1.27e3T + 4.93e5T^{2} \)
83 \( 1 + 493. iT - 5.71e5T^{2} \)
89 \( 1 - 1.25e3iT - 7.04e5T^{2} \)
97 \( 1 - 76.4iT - 9.12e5T^{2} \)
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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

−10.90280968520125063018307772707, −9.943597773007236546703561602294, −8.689274236990715968000008538054, −7.72479083116648241851206638836, −7.02967215292998769105223548908, −5.95131510626949823059577996147, −5.54171122366723548898401440801, −4.01511132327781864734512868344, −2.68855120303773070480852308924, −1.07222135825094982655384104508, 0.802023352556778800832928409682, 2.04364423733717798384418442882, 3.22046910008069795493939537444, 4.58488450295047817733272972385, 5.51502064819630389014326413340, 6.84736234541586475735463335190, 7.22472315428030690115558832370, 8.755345627021164091219602737082, 9.655296934915269416100303854080, 10.53551299326740807397162610586

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