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Properties

Label 2-507-39.5-c1-0-4
Degree $2$
Conductor $507$
Sign $0.0736 - 0.997i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (−0.540 − 0.540i)2-s + (0.0858 + 1.72i)3-s − 1.41i·4-s + (0.996 + 0.996i)5-s + (0.888 − 0.981i)6-s + (1.80 + 1.80i)7-s + (−1.84 + 1.84i)8-s + (−2.98 + 0.296i)9-s − 1.07i·10-s + (−3.35 + 3.35i)11-s + (2.44 − 0.121i)12-s − 1.94i·14-s + (−1.63 + 1.80i)15-s − 0.837·16-s + 5.80·17-s + (1.77 + 1.45i)18-s + ⋯
L(s)  = 1  + (−0.382 − 0.382i)2-s + (0.0495 + 0.998i)3-s − 0.708i·4-s + (0.445 + 0.445i)5-s + (0.362 − 0.400i)6-s + (0.681 + 0.681i)7-s + (−0.652 + 0.652i)8-s + (−0.995 + 0.0989i)9-s − 0.340i·10-s + (−1.01 + 1.01i)11-s + (0.707 − 0.0350i)12-s − 0.520i·14-s + (−0.422 + 0.467i)15-s − 0.209·16-s + 1.40·17-s + (0.417 + 0.342i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0736 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0736 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0736 - 0.997i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0736 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.787995 + 0.731921i\)
\(L(\frac12)\) \(\approx\) \(0.787995 + 0.731921i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0858 - 1.72i)T \)
13 \( 1 \)
good2 \( 1 + (0.540 + 0.540i)T + 2iT^{2} \)
5 \( 1 + (-0.996 - 0.996i)T + 5iT^{2} \)
7 \( 1 + (-1.80 - 1.80i)T + 7iT^{2} \)
11 \( 1 + (3.35 - 3.35i)T - 11iT^{2} \)
17 \( 1 - 5.80T + 17T^{2} \)
19 \( 1 + (2.39 - 2.39i)T - 19iT^{2} \)
23 \( 1 + 3.39T + 23T^{2} \)
29 \( 1 - 6.57iT - 29T^{2} \)
31 \( 1 + (0.386 - 0.386i)T - 31iT^{2} \)
37 \( 1 + (-5.93 - 5.93i)T + 37iT^{2} \)
41 \( 1 + (0.734 + 0.734i)T + 41iT^{2} \)
43 \( 1 - 7.56iT - 43T^{2} \)
47 \( 1 + (-0.243 + 0.243i)T - 47iT^{2} \)
53 \( 1 - 2.07iT - 53T^{2} \)
59 \( 1 + (3.56 - 3.56i)T - 59iT^{2} \)
61 \( 1 - 7.04T + 61T^{2} \)
67 \( 1 + (-4.54 + 4.54i)T - 67iT^{2} \)
71 \( 1 + (6.79 + 6.79i)T + 71iT^{2} \)
73 \( 1 + (-6.04 - 6.04i)T + 73iT^{2} \)
79 \( 1 - 8.77T + 79T^{2} \)
83 \( 1 + (8.31 + 8.31i)T + 83iT^{2} \)
89 \( 1 + (-9.62 + 9.62i)T - 89iT^{2} \)
97 \( 1 + (-1.34 + 1.34i)T - 97iT^{2} \)
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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

−10.79444371254273319893086034090, −10.09069630401630830700970479434, −9.798056300562217766067195148483, −8.649323122474220434667817545352, −7.84383224081159503928270009350, −6.18574379445520170896882035536, −5.40845413938908047879135216829, −4.65348651091447622478199383172, −2.93759930382338909971485111361, −1.93108673412243754568020864078, 0.71491655966636289917736167440, 2.41901246069312047508902490914, 3.72805124006273775740546284561, 5.32679928093137220724192564786, 6.18820325937606783469254170074, 7.41422121579503733210190282682, 7.923196833248429366137498781122, 8.523684377076861793719134593036, 9.585799990698051105209669595065, 10.81469910600148278638445135262

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