Properties

Label 2-507-39.11-c1-0-25
Degree $2$
Conductor $507$
Sign $0.0257 - 0.999i$
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  + (2.31 + 0.619i)2-s + (0.866 + 1.5i)3-s + (3.23 + 1.86i)4-s + (−1.23 + 1.23i)5-s + (1.07 + 4.00i)6-s + (2.93 + 2.93i)8-s + (−1.5 + 2.59i)9-s + (−3.63 + 2.09i)10-s + (1.69 − 6.31i)11-s + 6.46i·12-s + (−2.93 − 0.785i)15-s + (1.23 + 2.13i)16-s + (−5.07 + 5.07i)18-s + (−6.31 + 1.69i)20-s + (7.83 − 13.5i)22-s + ⋯
L(s)  = 1  + (1.63 + 0.438i)2-s + (0.499 + 0.866i)3-s + (1.61 + 0.933i)4-s + (−0.554 + 0.554i)5-s + (0.438 + 1.63i)6-s + (1.03 + 1.03i)8-s + (−0.5 + 0.866i)9-s + (−1.14 + 0.663i)10-s + (0.510 − 1.90i)11-s + 1.86i·12-s + (−0.757 − 0.202i)15-s + (0.308 + 0.533i)16-s + (−1.19 + 1.19i)18-s + (−1.41 + 0.378i)20-s + (1.66 − 2.89i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0257 - 0.999i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0257 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63480 + 2.56771i\)
\(L(\frac12)\) \(\approx\) \(2.63480 + 2.56771i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
13 \( 1 \)
good2 \( 1 + (-2.31 - 0.619i)T + (1.73 + i)T^{2} \)
5 \( 1 + (1.23 - 1.23i)T - 5iT^{2} \)
7 \( 1 + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.69 + 6.31i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-7.55 - 2.02i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.46 - 2i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.10 + 7.10i)T + 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-0.453 + 0.121i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.92 - 12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (4.17 + 15.5i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (8.91 - 8.91i)T - 83iT^{2} \)
89 \( 1 + (1.11 - 4.17i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (84.0 - 48.5i)T^{2} \)
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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

−11.28430140420076721228729271090, −10.65672153698792997598191768812, −9.242850461847932755091505336326, −8.285946222846588377076207628120, −7.31729298990015027633031830552, −6.17552889206538220450024168751, −5.43986292700352144443774142421, −4.24515812705287183770135197555, −3.51062001394689198702561074938, −2.83894836680697023088072432065, 1.61952475841461065483550852555, 2.69961439291851440483029780046, 4.03031172764171224342283785422, 4.64173271410630177290949721004, 5.92283952081654133871477847041, 6.91066093628545263928937303087, 7.65196693298823977001002321826, 8.865994799991644726523298652083, 9.913107181642176639511117609924, 11.24995224188582753994626894837

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