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Properties

Label 2-507-169.4-c1-0-1
Degree $2$
Conductor $507$
Sign $-0.975 - 0.222i$
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  + (−2.53 − 0.102i)2-s + (−0.278 + 0.960i)3-s + (4.44 + 0.358i)4-s + (1.66 + 0.629i)5-s + (0.804 − 2.41i)6-s + (−2.05 − 4.81i)7-s + (−6.20 − 0.753i)8-s + (−0.845 − 0.534i)9-s + (−4.15 − 1.76i)10-s + (0.626 + 0.990i)11-s + (−1.58 + 4.17i)12-s + (−3.48 + 0.936i)13-s + (4.71 + 12.4i)14-s + (−1.06 + 1.41i)15-s + (6.88 + 1.11i)16-s + (−4.86 + 2.07i)17-s + ⋯
L(s)  = 1  + (−1.79 − 0.0723i)2-s + (−0.160 + 0.554i)3-s + (2.22 + 0.179i)4-s + (0.742 + 0.281i)5-s + (0.328 − 0.984i)6-s + (−0.775 − 1.81i)7-s + (−2.19 − 0.266i)8-s + (−0.281 − 0.178i)9-s + (−1.31 − 0.559i)10-s + (0.188 + 0.298i)11-s + (−0.456 + 1.20i)12-s + (−0.965 + 0.259i)13-s + (1.26 + 3.32i)14-s + (−0.275 + 0.366i)15-s + (1.72 + 0.279i)16-s + (−1.17 + 0.502i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.975 - 0.222i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.975 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0100750 + 0.0895851i\)
\(L(\frac12)\) \(\approx\) \(0.0100750 + 0.0895851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + (3.48 - 0.936i)T \)
good2 \( 1 + (2.53 + 0.102i)T + (1.99 + 0.160i)T^{2} \)
5 \( 1 + (-1.66 - 0.629i)T + (3.74 + 3.31i)T^{2} \)
7 \( 1 + (2.05 + 4.81i)T + (-4.84 + 5.04i)T^{2} \)
11 \( 1 + (-0.626 - 0.990i)T + (-4.71 + 9.93i)T^{2} \)
17 \( 1 + (4.86 - 2.07i)T + (11.7 - 12.2i)T^{2} \)
19 \( 1 + (-0.350 + 0.202i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.21 - 2.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.322 - 8.00i)T + (-28.9 - 2.33i)T^{2} \)
31 \( 1 + (-3.37 - 3.80i)T + (-3.73 + 30.7i)T^{2} \)
37 \( 1 + (-1.51 + 0.309i)T + (34.0 - 14.5i)T^{2} \)
41 \( 1 + (9.64 + 2.79i)T + (34.6 + 21.9i)T^{2} \)
43 \( 1 + (1.80 - 8.85i)T + (-39.5 - 16.8i)T^{2} \)
47 \( 1 + (7.01 - 4.84i)T + (16.6 - 43.9i)T^{2} \)
53 \( 1 + (-0.532 + 4.38i)T + (-51.4 - 12.6i)T^{2} \)
59 \( 1 + (-0.739 - 4.55i)T + (-55.9 + 18.6i)T^{2} \)
61 \( 1 + (0.780 - 0.586i)T + (16.9 - 58.5i)T^{2} \)
67 \( 1 + (0.245 + 3.03i)T + (-66.1 + 10.7i)T^{2} \)
71 \( 1 + (-5.05 + 4.85i)T + (2.85 - 70.9i)T^{2} \)
73 \( 1 + (-1.42 + 2.70i)T + (-41.4 - 60.0i)T^{2} \)
79 \( 1 + (6.95 + 10.0i)T + (-28.0 + 73.8i)T^{2} \)
83 \( 1 + (-2.33 + 9.46i)T + (-73.4 - 38.5i)T^{2} \)
89 \( 1 + (14.6 + 8.47i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 8.95i)T + (19.4 + 95.0i)T^{2} \)
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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

−10.72734912915303876632123316347, −10.20758801881192464813523580074, −9.764853593313339812213868158000, −8.990476916995593976282602423201, −7.78209738519058929243319713124, −6.84815282806696069833857363247, −6.45889086247185437501696063931, −4.59135998501244970388062257708, −3.17106488792396038258064105018, −1.64889516535279752061395840204, 0.090127446630019628498889466138, 2.06380389028709035727884689809, 2.60617978875754033109245059941, 5.36138826140931430704521266324, 6.22896695708036411204452182947, 6.87089950692736547150485785667, 8.177437585847748389061945936749, 8.742110658070010289532269172913, 9.637635265914347333309175380558, 9.918967617153760463139319047622

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