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Properties

Label 2-507-13.12-c1-0-16
Degree $2$
Conductor $507$
Sign $-0.999 + 0.0304i$
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  − 2.04i·2-s − 3-s − 2.19·4-s + 3.35i·5-s + 2.04i·6-s − 2.24i·7-s + 0.405i·8-s + 9-s + 6.87·10-s − 4.93i·11-s + 2.19·12-s − 4.60·14-s − 3.35i·15-s − 3.56·16-s − 0.911·17-s − 2.04i·18-s + ⋯
L(s)  = 1  − 1.44i·2-s − 0.577·3-s − 1.09·4-s + 1.50i·5-s + 0.836i·6-s − 0.849i·7-s + 0.143i·8-s + 0.333·9-s + 2.17·10-s − 1.48i·11-s + 0.634·12-s − 1.23·14-s − 0.866i·15-s − 0.891·16-s − 0.221·17-s − 0.482i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.999 + 0.0304i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.999 + 0.0304i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0131194 - 0.860505i\)
\(L(\frac12)\) \(\approx\) \(0.0131194 - 0.860505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + 2.04iT - 2T^{2} \)
5 \( 1 - 3.35iT - 5T^{2} \)
7 \( 1 + 2.24iT - 7T^{2} \)
11 \( 1 + 4.93iT - 11T^{2} \)
17 \( 1 + 0.911T + 17T^{2} \)
19 \( 1 + 3.80iT - 19T^{2} \)
23 \( 1 + 2.02T + 23T^{2} \)
29 \( 1 + 3.93T + 29T^{2} \)
31 \( 1 + 8.82iT - 31T^{2} \)
37 \( 1 + 8.80iT - 37T^{2} \)
41 \( 1 - 6.93iT - 41T^{2} \)
43 \( 1 - 2.28T + 43T^{2} \)
47 \( 1 + 3.80iT - 47T^{2} \)
53 \( 1 - 0.542T + 53T^{2} \)
59 \( 1 - 4.71iT - 59T^{2} \)
61 \( 1 - 3.67T + 61T^{2} \)
67 \( 1 + 1.52iT - 67T^{2} \)
71 \( 1 - 2.37iT - 71T^{2} \)
73 \( 1 + 7.41iT - 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 - 2.30iT - 83T^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 + 16.1iT - 97T^{2} \)
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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

−10.75905542716570329889018349773, −10.10907138966350564311156594887, −9.125862947781322412477327224224, −7.63769959314459092534406727455, −6.75242783375162512087516102689, −5.84536432878628241158907586900, −4.17382617570952520916689324090, −3.39744102875102796300558873916, −2.35604807238106439564765285765, −0.54673112717218748091963727800, 1.81395735862118071612012367370, 4.33565236143602620048484189321, 5.09253788230750429277287069831, 5.68280000537407585093511917077, 6.72179353779546891002289363876, 7.68614619826885888584460994336, 8.559344417811551184445240956430, 9.215039591688014873623507170761, 10.15737885719385960237573866565, 11.66360596193562793981113548314

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