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Properties

Label 2-507-13.12-c1-0-15
Degree $2$
Conductor $507$
Sign $0.832 - 0.554i$
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  + 2.41i·2-s + 3-s − 3.82·4-s − 2.82i·5-s + 2.41i·6-s − 2.82i·7-s − 4.41i·8-s + 9-s + 6.82·10-s − 2i·11-s − 3.82·12-s + 6.82·14-s − 2.82i·15-s + 2.99·16-s + 3.65·17-s + 2.41i·18-s + ⋯
L(s)  = 1  + 1.70i·2-s + 0.577·3-s − 1.91·4-s − 1.26i·5-s + 0.985i·6-s − 1.06i·7-s − 1.56i·8-s + 0.333·9-s + 2.15·10-s − 0.603i·11-s − 1.10·12-s + 1.82·14-s − 0.730i·15-s + 0.749·16-s + 0.886·17-s + 0.569i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.832 - 0.554i$
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: \((2,\ 507,\ (\ :1/2),\ 0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45288 + 0.439897i\)
\(L(\frac12)\) \(\approx\) \(1.45288 + 0.439897i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41iT - 2T^{2} \)
5 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 6.82iT - 31T^{2} \)
37 \( 1 - 3.65iT - 37T^{2} \)
41 \( 1 + 10.8iT - 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 + 0.343iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 3.65iT - 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 + 1.17iT - 67T^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 - 11.6iT - 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 7.65iT - 83T^{2} \)
89 \( 1 - 9.17iT - 89T^{2} \)
97 \( 1 - 7.65iT - 97T^{2} \)
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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

−10.73780780353421547165672188163, −9.622218269578828416285685392731, −8.780134153246329571338444612932, −8.282427084307224748968113301023, −7.38483113781490840572158275796, −6.62462002213000556669800919970, −5.30581934057749189589258873043, −4.71066652253955467134733220451, −3.54308177137946858808384954457, −0.935288527558163476420023430515, 1.80791716948160447606819727956, 2.79596718281181858588101229083, 3.38869672699341469396418259357, 4.70352466768922871050771490419, 6.10226728422057323485812251585, 7.39849352577358099294745336808, 8.473076031815888671320834460009, 9.498651528362319813662701871113, 10.00091177361788799399358362421, 10.84518768479062398881798214587

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