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Properties

Label 2-507-507.422-c1-0-43
Degree $2$
Conductor $507$
Sign $-0.898 + 0.438i$
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  + (−1.38 − 1.04i)3-s + (−1.99 + 0.0805i)4-s + (3.92 − 2.59i)7-s + (0.834 + 2.88i)9-s + (2.85 + 1.96i)12-s + (−2.59 − 2.5i)13-s + (3.98 − 0.321i)16-s + (−8.34 − 2.23i)19-s + (−8.13 − 0.492i)21-s + (−4.67 + 1.77i)25-s + (1.84 − 4.85i)27-s + (−7.63 + 5.50i)28-s + (−3.80 − 8.44i)31-s + (−1.90 − 5.69i)36-s + (−0.634 + 6.27i)37-s + ⋯
L(s)  = 1  + (−0.799 − 0.600i)3-s + (−0.999 + 0.0402i)4-s + (1.48 − 0.980i)7-s + (0.278 + 0.960i)9-s + (0.822 + 0.568i)12-s + (−0.720 − 0.693i)13-s + (0.996 − 0.0804i)16-s + (−1.91 − 0.512i)19-s + (−1.77 − 0.107i)21-s + (−0.935 + 0.354i)25-s + (0.354 − 0.935i)27-s + (−1.44 + 1.03i)28-s + (−0.682 − 1.51i)31-s + (−0.316 − 0.948i)36-s + (−0.104 + 1.03i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.898 + 0.438i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.898 + 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123466 - 0.534981i\)
\(L(\frac12)\) \(\approx\) \(0.123466 - 0.534981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.38 + 1.04i)T \)
13 \( 1 + (2.59 + 2.5i)T \)
good2 \( 1 + (1.99 - 0.0805i)T^{2} \)
5 \( 1 + (4.67 - 1.77i)T^{2} \)
7 \( 1 + (-3.92 + 2.59i)T + (2.74 - 6.43i)T^{2} \)
11 \( 1 + (-5.87 + 9.29i)T^{2} \)
17 \( 1 + (-15.6 - 6.66i)T^{2} \)
19 \( 1 + (8.34 + 2.23i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.16 + 28.9i)T^{2} \)
31 \( 1 + (3.80 + 8.44i)T + (-20.5 + 23.2i)T^{2} \)
37 \( 1 + (0.634 - 6.27i)T + (-36.2 - 7.40i)T^{2} \)
41 \( 1 + (-39.3 + 11.4i)T^{2} \)
43 \( 1 + (7.68 + 6.27i)T + (8.60 + 42.1i)T^{2} \)
47 \( 1 + (-38.6 - 26.6i)T^{2} \)
53 \( 1 + (-6.38 - 52.6i)T^{2} \)
59 \( 1 + (9.46 - 58.2i)T^{2} \)
61 \( 1 + (1.54 + 0.515i)T + (48.7 + 36.6i)T^{2} \)
67 \( 1 + (-10.1 - 11.0i)T + (-5.39 + 66.7i)T^{2} \)
71 \( 1 + (51.2 + 49.1i)T^{2} \)
73 \( 1 + (1.16 + 0.703i)T + (33.9 + 64.6i)T^{2} \)
79 \( 1 + (-12.7 + 6.67i)T + (44.8 - 65.0i)T^{2} \)
83 \( 1 + (-19.8 - 80.5i)T^{2} \)
89 \( 1 + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.70 + 16.0i)T + (-75.1 + 61.3i)T^{2} \)
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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

−10.63849344980434465305174468043, −9.865623246260261075358960388329, −8.407375130455621647546470234258, −7.902138644662107560296296780448, −7.00154811239253190792538680870, −5.65232557093239521062181237753, −4.80836290886011943744455010652, −4.10839430256648580347510145180, −1.90017575046969038539116888476, −0.37229277668636410499864551311, 1.88275293783439259555165178819, 3.95477909895066564657953174747, 4.78267483656938492560124179511, 5.37644044558248036075826160959, 6.44483470467750214075709845053, 7.971358582321409939874805511130, 8.739652227896168466100075344408, 9.462513734146552314777600173255, 10.47847611250046306577730097822, 11.21976963903389052374266427011

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