Properties

Label 2-513-171.128-c1-0-9
Degree $2$
Conductor $513$
Sign $0.630 - 0.776i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
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  + (−0.114 + 0.651i)2-s + (1.46 + 0.534i)4-s + (2.11 + 0.373i)5-s + 4.63·7-s + (−1.17 + 2.04i)8-s + (−0.486 + 1.33i)10-s + (−2.06 + 1.19i)11-s + (−1.12 − 1.34i)13-s + (−0.532 + 3.02i)14-s + (1.19 + 1.00i)16-s + (−1.52 − 4.20i)17-s + (−3.09 − 3.06i)19-s + (2.90 + 1.67i)20-s + (−0.541 − 1.48i)22-s + (1.97 − 5.41i)23-s + ⋯
L(s)  = 1  + (−0.0812 + 0.460i)2-s + (0.733 + 0.267i)4-s + (0.946 + 0.166i)5-s + 1.75·7-s + (−0.416 + 0.721i)8-s + (−0.153 + 0.422i)10-s + (−0.623 + 0.360i)11-s + (−0.312 − 0.371i)13-s + (−0.142 + 0.807i)14-s + (0.299 + 0.251i)16-s + (−0.371 − 1.01i)17-s + (−0.710 − 0.703i)19-s + (0.650 + 0.375i)20-s + (−0.115 − 0.316i)22-s + (0.411 − 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.630 - 0.776i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.630 - 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88066 + 0.895611i\)
\(L(\frac12)\) \(\approx\) \(1.88066 + 0.895611i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (3.09 + 3.06i)T \)
good2 \( 1 + (0.114 - 0.651i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (-2.11 - 0.373i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 - 4.63T + 7T^{2} \)
11 \( 1 + (2.06 - 1.19i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.12 + 1.34i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.52 + 4.20i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-1.97 + 5.41i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (5.01 - 4.21i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.62 + 1.51i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.49iT - 37T^{2} \)
41 \( 1 + (-0.835 + 0.303i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (10.1 - 3.70i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-5.36 - 6.39i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-5.26 + 4.41i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (4.79 + 4.02i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.265 - 1.50i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.99 - 0.528i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-7.57 - 6.35i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.89 + 10.7i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.38 - 1.65i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 - 2.14iT - 83T^{2} \)
89 \( 1 + (2.28 + 12.9i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-1.81 - 0.320i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.01805723725311682196000809407, −10.32537330602170041775899564333, −9.055415225615415011625207475833, −8.175229103980671448269307865946, −7.41090888973035144286901985812, −6.54917644940149603399133269763, −5.36940654759902422718800136963, −4.74305773215246838612422066958, −2.67706440117078107603121013619, −1.89397226869144156612359491839, 1.66695573838931409343893867264, 2.13508968606115347795863644790, 3.91368251283720475650878921177, 5.34387965054427496442190835280, 5.87930581131902098826585483000, 7.20720851936480332754334873853, 8.105502000033489812446917984199, 9.091695912291709156803999372685, 10.14234842491930973644606273285, 10.82704527551679979162132075822

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