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Properties

Label 2-507-13.3-c1-0-2
Degree $2$
Conductor $507$
Sign $0.522 + 0.852i$
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  + (−1.20 − 2.09i)2-s + (−0.5 − 0.866i)3-s + (−1.91 + 3.31i)4-s − 2.82·5-s + (−1.20 + 2.09i)6-s + (−1.41 + 2.44i)7-s + 4.41·8-s + (−0.499 + 0.866i)9-s + (3.41 + 5.91i)10-s + (−1 − 1.73i)11-s + 3.82·12-s + 6.82·14-s + (1.41 + 2.44i)15-s + (−1.49 − 2.59i)16-s + (1.82 − 3.16i)17-s + 2.41·18-s + ⋯
L(s)  = 1  + (−0.853 − 1.47i)2-s + (−0.288 − 0.499i)3-s + (−0.957 + 1.65i)4-s − 1.26·5-s + (−0.492 + 0.853i)6-s + (−0.534 + 0.925i)7-s + 1.56·8-s + (−0.166 + 0.288i)9-s + (1.07 + 1.87i)10-s + (−0.301 − 0.522i)11-s + 1.10·12-s + 1.82·14-s + (0.365 + 0.632i)15-s + (−0.374 − 0.649i)16-s + (0.443 − 0.768i)17-s + 0.569·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.368101 - 0.206275i\)
\(L(\frac12)\) \(\approx\) \(0.368101 - 0.206275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (1.20 + 2.09i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
7 \( 1 + (1.41 - 2.44i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.82 + 3.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + (-1.82 - 3.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.41 - 9.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.82 - 8.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.343T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (1.82 - 3.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.65 + 8.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.585 - 1.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1 + 1.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 7.65T + 83T^{2} \)
89 \( 1 + (-4.58 - 7.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.82 - 6.63i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.13222969888195980429462597313, −9.883618923950290099840161393058, −9.173615530531791896315117298857, −8.189742476520684086030066841045, −7.63766527649877479335388198009, −6.24109139264742190503335946828, −4.79152638643166828516196947337, −3.33327621237102495705815664279, −2.68088528151026880490290176226, −0.878569479498000340097638638849, 0.51156222768782491682817702019, 3.61134353718673147896807893578, 4.53521804362827743023511464805, 5.69883922840257998416938983799, 6.78441309465247104042041703103, 7.44887170621006871566221752535, 8.158344428701247027218031688698, 9.053173692677022045589583625512, 10.17901611257579890689381422858, 10.51147240422101642633235870186

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