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Properties

Label 2-507-13.3-c1-0-23
Degree $2$
Conductor $507$
Sign $-0.990 + 0.134i$
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  + (−0.222 − 0.385i)2-s + (−0.5 − 0.866i)3-s + (0.900 − 1.56i)4-s − 0.246·5-s + (−0.222 + 0.385i)6-s + (−0.876 + 1.51i)7-s − 1.69·8-s + (−0.499 + 0.866i)9-s + (0.0549 + 0.0951i)10-s + (−2.82 − 4.89i)11-s − 1.80·12-s + 0.780·14-s + (0.123 + 0.213i)15-s + (−1.42 − 2.46i)16-s + (1.90 − 3.29i)17-s + 0.445·18-s + ⋯
L(s)  = 1  + (−0.157 − 0.272i)2-s + (−0.288 − 0.499i)3-s + (0.450 − 0.780i)4-s − 0.110·5-s + (−0.0908 + 0.157i)6-s + (−0.331 + 0.573i)7-s − 0.598·8-s + (−0.166 + 0.288i)9-s + (0.0173 + 0.0301i)10-s + (−0.852 − 1.47i)11-s − 0.520·12-s + 0.208·14-s + (0.0318 + 0.0552i)15-s + (−0.356 − 0.617i)16-s + (0.461 − 0.798i)17-s + 0.104·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.990 + 0.134i$
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: \((2,\ 507,\ (\ :1/2),\ -0.990 + 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0504093 - 0.748358i\)
\(L(\frac12)\) \(\approx\) \(0.0504093 - 0.748358i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.222 + 0.385i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.246T + 5T^{2} \)
7 \( 1 + (0.876 - 1.51i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.82 + 4.89i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.90 + 3.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.79 - 4.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.17 + 7.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.96 - 5.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.26T + 31T^{2} \)
37 \( 1 + (1.59 + 2.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.222 + 0.385i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.856 - 1.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.73T + 47T^{2} \)
53 \( 1 + 1.06T + 53T^{2} \)
59 \( 1 + (-6.85 + 11.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.25 + 7.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.98 + 5.16i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.85 + 4.95i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.35T + 73T^{2} \)
79 \( 1 - 4.45T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (0.0685 + 0.118i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.84 + 11.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.53438284102927230206595023201, −9.853702969945341046877987159111, −8.609435034489859720257466549245, −7.915882804255899464095206084156, −6.47904363128345628248690065842, −5.99439522831110963680160335571, −5.08181199096288147535901230731, −3.21219233542648758830026458624, −2.14359875952005769838066149262, −0.44972993071087447733884193620, 2.30910479003689922645922518991, 3.67913489503066777145453480677, 4.54938569204671455149643794183, 5.88736993533440203811861275889, 6.93356622761945831398744815752, 7.65733193357598676726824760097, 8.492280638687950014856604151555, 9.861820254435301790743607111453, 10.18808547476277163348091464927, 11.43662107407081152881030478276

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