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Properties

Label 2-507-13.12-c1-0-19
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  + 0.445i·2-s + 3-s + 1.80·4-s − 0.246i·5-s + 0.445i·6-s − 1.75i·7-s + 1.69i·8-s + 9-s + 0.109·10-s − 5.65i·11-s + 1.80·12-s + 0.780·14-s − 0.246i·15-s + 2.85·16-s + 3.80·17-s + 0.445i·18-s + ⋯
L(s)  = 1  + 0.314i·2-s + 0.577·3-s + 0.900·4-s − 0.110i·5-s + 0.181i·6-s − 0.662i·7-s + 0.598i·8-s + 0.333·9-s + 0.0347·10-s − 1.70i·11-s + 0.520·12-s + 0.208·14-s − 0.0637i·15-s + 0.712·16-s + 0.922·17-s + 0.104i·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

\(L(1)\) \(\approx\) \(2.13315 - 0.0325224i\)
\(L(\frac12)\) \(\approx\) \(2.13315 - 0.0325224i\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.445iT - 2T^{2} \)
5 \( 1 + 0.246iT - 5T^{2} \)
7 \( 1 + 1.75iT - 7T^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
17 \( 1 - 3.80T + 17T^{2} \)
19 \( 1 - 5.58iT - 19T^{2} \)
23 \( 1 + 8.34T + 23T^{2} \)
29 \( 1 + 5.93T + 29T^{2} \)
31 \( 1 - 5.26iT - 31T^{2} \)
37 \( 1 + 3.19iT - 37T^{2} \)
41 \( 1 - 0.445iT - 41T^{2} \)
43 \( 1 + 1.71T + 43T^{2} \)
47 \( 1 - 6.73iT - 47T^{2} \)
53 \( 1 + 1.06T + 53T^{2} \)
59 \( 1 - 13.7iT - 59T^{2} \)
61 \( 1 + 8.51T + 61T^{2} \)
67 \( 1 - 5.96iT - 67T^{2} \)
71 \( 1 + 5.71iT - 71T^{2} \)
73 \( 1 + 7.35iT - 73T^{2} \)
79 \( 1 - 4.45T + 79T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 + 0.137iT - 89T^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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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

−10.72412481497597963405747745494, −10.24539457777844169022858539027, −8.921294905727026955640948686275, −8.004202308045549078512846833375, −7.50572564955301573772165105199, −6.24766872631425132870450504561, −5.59350790785620267174477527742, −3.86212634835838989340430360574, −3.02789763394512717093033262098, −1.44473582901668101392635924908, 1.86597989438377714915301595061, 2.64290923768552615752740368747, 3.92448575173771854660425022154, 5.24178115727852240199357477249, 6.50632841522595766178616325589, 7.31743384826727686980791968289, 8.099742919823764243089735286094, 9.430312392122286803725456284925, 9.908882641553326456004333816061, 10.90728128734156972314831248425

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