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Properties

Label 2-507-39.2-c1-0-16
Degree $2$
Conductor $507$
Sign $0.176 + 0.984i$
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  + (0.389 − 1.45i)2-s + (−1.60 − 0.650i)3-s + (−0.232 − 0.133i)4-s + (1.06 + 1.06i)5-s + (−1.57 + 2.08i)6-s + (−1.36 + 0.366i)7-s + (1.84 − 1.84i)8-s + (2.15 + 2.08i)9-s + (1.96 − 1.13i)10-s + (3.97 + 1.06i)11-s + (0.285 + 0.366i)12-s + 2.12i·14-s + (−1.01 − 2.40i)15-s + (−2.23 − 3.86i)16-s + (2.51 − 4.36i)17-s + (3.87 − 2.31i)18-s + ⋯
L(s)  = 1  + (0.275 − 1.02i)2-s + (−0.926 − 0.375i)3-s + (−0.116 − 0.0669i)4-s + (0.476 + 0.476i)5-s + (−0.641 + 0.849i)6-s + (−0.516 + 0.138i)7-s + (0.652 − 0.652i)8-s + (0.717 + 0.696i)9-s + (0.621 − 0.358i)10-s + (1.19 + 0.321i)11-s + (0.0823 + 0.105i)12-s + 0.569i·14-s + (−0.262 − 0.620i)15-s + (−0.558 − 0.966i)16-s + (0.611 − 1.05i)17-s + (0.913 − 0.546i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.176 + 0.984i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.176 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18735 - 0.993653i\)
\(L(\frac12)\) \(\approx\) \(1.18735 - 0.993653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 + 0.650i)T \)
13 \( 1 \)
good2 \( 1 + (-0.389 + 1.45i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.06 - 1.06i)T + 5iT^{2} \)
7 \( 1 + (1.36 - 0.366i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.97 - 1.06i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.51 + 4.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 3.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.20 + 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.46 - 2.46i)T - 31iT^{2} \)
37 \( 1 + (-1.40 + 5.23i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.45 - 5.42i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.90 + 1.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.25 - 4.25i)T - 47iT^{2} \)
53 \( 1 - 0.779iT - 53T^{2} \)
59 \( 1 + (-0.779 - 2.90i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.73 - 1.53i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.90 + 0.779i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.901 + 0.901i)T + 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (-2.90 - 2.90i)T + 83iT^{2} \)
89 \( 1 + (-9.01 - 2.41i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.437 + 1.63i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.87500412908219621609281831196, −10.00293422110625176226523351511, −9.543944333889390637777377634836, −7.79114302452437743385682854857, −6.76571768873209921653740352597, −6.23563398270310235547344306311, −4.90412029892720116310703188485, −3.72138929723535208912562362356, −2.48545379135997897526589580090, −1.21629463269790149575766129930, 1.37394573752243647868954203133, 3.67302046496477400316739704335, 4.84802398974841826296848015130, 5.63711520436726737110929256401, 6.45787779430270126088999341766, 6.97196832546008132320220510297, 8.368419181126935860283645891672, 9.344534087900956631294086398357, 10.19980148789751922171292001877, 11.11352156718148436497448052451

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