Properties

Label 2-507-13.3-c1-0-19
Degree $2$
Conductor $507$
Sign $-0.872 + 0.488i$
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  + (−0.780 − 1.35i)2-s + (−0.5 − 0.866i)3-s + (−0.219 + 0.379i)4-s + 3.56·5-s + (−0.780 + 1.35i)6-s + (0.280 − 0.486i)7-s − 2.43·8-s + (−0.499 + 0.866i)9-s + (−2.78 − 4.81i)10-s + (−1 − 1.73i)11-s + 0.438·12-s − 0.876·14-s + (−1.78 − 3.08i)15-s + (2.34 + 4.05i)16-s + (0.780 − 1.35i)17-s + 1.56·18-s + ⋯
L(s)  = 1  + (−0.552 − 0.956i)2-s + (−0.288 − 0.499i)3-s + (−0.109 + 0.189i)4-s + 1.59·5-s + (−0.318 + 0.552i)6-s + (0.106 − 0.183i)7-s − 0.862·8-s + (−0.166 + 0.288i)9-s + (−0.879 − 1.52i)10-s + (−0.301 − 0.522i)11-s + 0.126·12-s − 0.234·14-s + (−0.459 − 0.796i)15-s + (0.585 + 1.01i)16-s + (0.189 − 0.327i)17-s + 0.368·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.872 + 0.488i$
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: \((2,\ 507,\ (\ :1/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303864 - 1.16383i\)
\(L(\frac12)\) \(\approx\) \(0.303864 - 1.16383i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.780 + 1.35i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 3.56T + 5T^{2} \)
7 \( 1 + (-0.280 + 0.486i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.780 + 1.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.56 + 6.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.34 + 5.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.56T + 31T^{2} \)
37 \( 1 + (-3.78 - 6.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.780 + 1.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.28 - 3.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 + 0.684T + 53T^{2} \)
59 \( 1 + (1.43 - 2.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.93 - 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.28 - 3.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7 + 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 5.43T + 79T^{2} \)
83 \( 1 - 0.876T + 83T^{2} \)
89 \( 1 + (-2.43 - 4.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.28 - 7.41i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.59166399526128512603339041308, −9.649124351971963655876216382090, −9.246886646932221049148530385469, −8.024859747302351842951245503901, −6.68061807172844379866689607232, −5.93630573018603626252572068759, −5.04955947583767624716138085244, −2.98623095044222681985502288703, −2.11054503878639634575854936905, −0.908229563814700896464000294106, 1.90846095971533647900261052386, 3.45110746090661217016964648624, 5.28989216872990357745754169094, 5.70555074024199559945781498270, 6.63576926055383295530009825877, 7.64651741656644496436944365778, 8.665724483267901869657165252036, 9.630125900394697373951662510743, 9.910217737095316981207919228059, 11.02714778780201579775249673847

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