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Properties

Label 2-507-13.4-c1-0-16
Degree $2$
Conductor $507$
Sign $0.207 - 0.978i$
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  + (2.33 + 1.34i)2-s + (0.5 − 0.866i)3-s + (2.62 + 4.54i)4-s + 1.04i·5-s + (2.33 − 1.34i)6-s + (0.480 − 0.277i)7-s + 8.74i·8-s + (−0.499 − 0.866i)9-s + (−1.41 + 2.44i)10-s + (−2.52 − 1.45i)11-s + 5.24·12-s + 1.49·14-s + (0.908 + 0.524i)15-s + (−6.51 + 11.2i)16-s + (−2.42 − 4.20i)17-s − 2.69i·18-s + ⋯
L(s)  = 1  + (1.64 + 0.951i)2-s + (0.288 − 0.499i)3-s + (1.31 + 2.27i)4-s + 0.469i·5-s + (0.951 − 0.549i)6-s + (0.181 − 0.104i)7-s + 3.09i·8-s + (−0.166 − 0.288i)9-s + (−0.446 + 0.773i)10-s + (−0.760 − 0.438i)11-s + 1.51·12-s + 0.399·14-s + (0.234 + 0.135i)15-s + (−1.62 + 2.82i)16-s + (−0.588 − 1.01i)17-s − 0.634i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.207 - 0.978i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.207 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.89499 + 2.34459i\)
\(L(\frac12)\) \(\approx\) \(2.89499 + 2.34459i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.33 - 1.34i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 1.04iT - 5T^{2} \)
7 \( 1 + (-0.480 + 0.277i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.52 + 1.45i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.42 + 4.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.652 + 0.376i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.88 + 4.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.955 + 1.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.51iT - 31T^{2} \)
37 \( 1 + (4.98 + 2.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.25 - 2.45i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.54 + 9.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.753iT - 47T^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 + (-3.54 + 2.04i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.71 - 2.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.62 + 0.936i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.09 + 5.25i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 + 2.64iT - 83T^{2} \)
89 \( 1 + (8.59 + 4.96i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.7 - 8.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.38065656602882235820319194386, −10.63897895522825877063673513986, −8.841070983663589179631069431152, −8.027666461688483622496147594323, −7.01481532686887812052971366698, −6.66493875808861979659844425840, −5.41241600228626661062015795596, −4.66618351808117958863478030394, −3.27660488809077523792526334657, −2.55823263129670432159288343071, 1.71316389247006630785094734904, 2.88975012722064662297209956211, 3.95902523575848239824068050163, 4.84183736804509358903272645850, 5.49111033434595110141036457608, 6.63925270211131829514398128802, 8.029457822741368645934044217938, 9.335570651765071642378800815012, 10.15378346004200426029331182176, 10.97351053515211966544772497488

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