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Properties

Label 2-507-39.2-c1-0-18
Degree $2$
Conductor $507$
Sign $0.533 + 0.846i$
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  + (0.619 − 2.31i)2-s + (0.866 + 1.5i)3-s + (−3.23 − 1.86i)4-s + (1.23 + 1.23i)5-s + (4.00 − 1.07i)6-s + (−2.93 + 2.93i)8-s + (−1.5 + 2.59i)9-s + (3.63 − 2.09i)10-s + (6.31 + 1.69i)11-s − 6.46i·12-s + (−0.785 + 2.93i)15-s + (1.23 + 2.13i)16-s + (5.07 + 5.07i)18-s + (−1.69 − 6.31i)20-s + (7.83 − 13.5i)22-s + ⋯
L(s)  = 1  + (0.438 − 1.63i)2-s + (0.499 + 0.866i)3-s + (−1.61 − 0.933i)4-s + (0.554 + 0.554i)5-s + (1.63 − 0.438i)6-s + (−1.03 + 1.03i)8-s + (−0.5 + 0.866i)9-s + (1.14 − 0.663i)10-s + (1.90 + 0.510i)11-s − 1.86i·12-s + (−0.202 + 0.757i)15-s + (0.308 + 0.533i)16-s + (1.19 + 1.19i)18-s + (−0.378 − 1.41i)20-s + (1.66 − 2.89i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.533 + 0.846i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.533 + 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94765 - 1.07488i\)
\(L(\frac12)\) \(\approx\) \(1.94765 - 1.07488i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.619 + 2.31i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.23 - 1.23i)T + 5iT^{2} \)
7 \( 1 + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-6.31 - 1.69i)T + (9.52 + 5.5i)T^{2} \)
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 \( 1 + (-2.02 + 7.55i)T + (-35.5 - 20.5i)T^{2} \)
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 \( 1 + (-0.121 - 0.453i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.92 - 12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (15.5 - 4.17i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (-8.91 - 8.91i)T + 83iT^{2} \)
89 \( 1 + (4.17 + 1.11i)T + (77.0 + 44.5i)T^{2} \)
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

−10.66675318433464831092975500659, −10.09185580617744469236840519063, −9.357224695980054375459852789892, −8.747841097633110119903713886069, −7.04774918222428956665985648111, −5.75121170600826549695010430161, −4.45430129627352214445097803791, −3.83117910821008392603431391658, −2.76756259704859255798247468964, −1.69938318644832911656625623526, 1.43735324962412166409103779611, 3.46883369951088156917398547757, 4.63551374383456508573874680378, 5.93736053160599651970331953924, 6.37152749939267286425864190600, 7.26396410043501183134563452619, 8.194930060343153048645136338328, 8.996296957172584094836810357017, 9.450604904442609960410433291819, 11.35287882473406867547404732002

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