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Properties

Label 2-507-13.12-c3-0-57
Degree $2$
Conductor $507$
Sign $0.832 + 0.554i$
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  + i·2-s + 3·3-s + 7·4-s − 7i·5-s + 3i·6-s − 10i·7-s + 15i·8-s + 9·9-s + 7·10-s − 22i·11-s + 21·12-s + 10·14-s − 21i·15-s + 41·16-s − 37·17-s + 9i·18-s + ⋯
L(s)  = 1  + 0.353i·2-s + 0.577·3-s + 0.875·4-s − 0.626i·5-s + 0.204i·6-s − 0.539i·7-s + 0.662i·8-s + 0.333·9-s + 0.221·10-s − 0.603i·11-s + 0.505·12-s + 0.190·14-s − 0.361i·15-s + 0.640·16-s − 0.527·17-s + 0.117i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.832 + 0.554i$
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: \((2,\ 507,\ (\ :3/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.073958774\)
\(L(\frac12)\) \(\approx\) \(3.073958774\)
\(L(\frac{5}{2})\) 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 \( 1 - iT - 8T^{2} \)
5 \( 1 + 7iT - 125T^{2} \)
7 \( 1 + 10iT - 343T^{2} \)
11 \( 1 + 22iT - 1.33e3T^{2} \)
17 \( 1 + 37T + 4.91e3T^{2} \)
19 \( 1 + 30iT - 6.85e3T^{2} \)
23 \( 1 - 162T + 1.21e4T^{2} \)
29 \( 1 + 113T + 2.43e4T^{2} \)
31 \( 1 + 196iT - 2.97e4T^{2} \)
37 \( 1 - 13iT - 5.06e4T^{2} \)
41 \( 1 + 285iT - 6.89e4T^{2} \)
43 \( 1 - 246T + 7.95e4T^{2} \)
47 \( 1 + 462iT - 1.03e5T^{2} \)
53 \( 1 + 537T + 1.48e5T^{2} \)
59 \( 1 - 576iT - 2.05e5T^{2} \)
61 \( 1 + 635T + 2.26e5T^{2} \)
67 \( 1 + 202iT - 3.00e5T^{2} \)
71 \( 1 - 1.08e3iT - 3.57e5T^{2} \)
73 \( 1 + 805iT - 3.89e5T^{2} \)
79 \( 1 - 884T + 4.93e5T^{2} \)
83 \( 1 + 518iT - 5.71e5T^{2} \)
89 \( 1 - 194iT - 7.04e5T^{2} \)
97 \( 1 - 1.20e3iT - 9.12e5T^{2} \)
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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

−10.60112324143922350474529930907, −9.259996534925932011471982368967, −8.626010499916792556473548319619, −7.59288940331421725512631197431, −6.95098959712692682577528235478, −5.84731868777668843860899627645, −4.74329879418714130111604751333, −3.46349867047418674221993607443, −2.29676760800215134062938286735, −0.905421738761332474586948828172, 1.50429643173402262795720732483, 2.61569970950581122404835494376, 3.31624187138806325052107449747, 4.77352743291811944145030170546, 6.19121548472192415221577139489, 6.98619977537798814530031605912, 7.73828231743449996037902114865, 8.949728984531394534499386743901, 9.728178961169051786925102325261, 10.79460748587518372048437793175

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