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Properties

Label 2-507-507.2-c1-0-38
Degree $2$
Conductor $507$
Sign $0.898 + 0.438i$
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  + (1.38 − 1.04i)3-s + (1.99 + 0.0805i)4-s + (−1.33 + 2.01i)7-s + (0.834 − 2.88i)9-s + (2.85 − 1.96i)12-s + (2.59 − 2.5i)13-s + (3.98 + 0.321i)16-s + (0.303 + 1.13i)19-s + (0.252 + 4.17i)21-s + (4.67 + 1.77i)25-s + (−1.84 − 4.85i)27-s + (−2.82 + 3.92i)28-s + (−5.64 − 2.53i)31-s + (1.90 − 5.69i)36-s + (−10.3 + 1.04i)37-s + ⋯
L(s)  = 1  + (0.799 − 0.600i)3-s + (0.999 + 0.0402i)4-s + (−0.503 + 0.762i)7-s + (0.278 − 0.960i)9-s + (0.822 − 0.568i)12-s + (0.720 − 0.693i)13-s + (0.996 + 0.0804i)16-s + (0.0697 + 0.260i)19-s + (0.0551 + 0.912i)21-s + (0.935 + 0.354i)25-s + (−0.354 − 0.935i)27-s + (−0.534 + 0.741i)28-s + (−1.01 − 0.455i)31-s + (0.316 − 0.948i)36-s + (−1.70 + 0.171i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.898 + 0.438i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.898 + 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18207 - 0.503594i\)
\(L(\frac12)\) \(\approx\) \(2.18207 - 0.503594i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.38 + 1.04i)T \)
13 \( 1 + (-2.59 + 2.5i)T \)
good2 \( 1 + (-1.99 - 0.0805i)T^{2} \)
5 \( 1 + (-4.67 - 1.77i)T^{2} \)
7 \( 1 + (1.33 - 2.01i)T + (-2.74 - 6.43i)T^{2} \)
11 \( 1 + (5.87 + 9.29i)T^{2} \)
17 \( 1 + (-15.6 + 6.66i)T^{2} \)
19 \( 1 + (-0.303 - 1.13i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.16 - 28.9i)T^{2} \)
31 \( 1 + (5.64 + 2.53i)T + (20.5 + 23.2i)T^{2} \)
37 \( 1 + (10.3 - 1.04i)T + (36.2 - 7.40i)T^{2} \)
41 \( 1 + (39.3 + 11.4i)T^{2} \)
43 \( 1 + (7.68 - 6.27i)T + (8.60 - 42.1i)T^{2} \)
47 \( 1 + (38.6 - 26.6i)T^{2} \)
53 \( 1 + (-6.38 + 52.6i)T^{2} \)
59 \( 1 + (-9.46 - 58.2i)T^{2} \)
61 \( 1 + (-1.54 + 0.515i)T + (48.7 - 36.6i)T^{2} \)
67 \( 1 + (4.80 + 4.43i)T + (5.39 + 66.7i)T^{2} \)
71 \( 1 + (-51.2 + 49.1i)T^{2} \)
73 \( 1 + (-8.81 - 14.5i)T + (-33.9 + 64.6i)T^{2} \)
79 \( 1 + (12.7 + 6.67i)T + (44.8 + 65.0i)T^{2} \)
83 \( 1 + (19.8 - 80.5i)T^{2} \)
89 \( 1 + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.36 + 3.33i)T + (75.1 + 61.3i)T^{2} \)
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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

−10.90396744431293251954600970485, −9.917439807574412758120195846798, −8.890457955065006246307471447759, −8.154702022336043191278244867919, −7.18637074692037028060970964157, −6.40245602425012624327640570369, −5.51835170735280411323587410494, −3.53542080349395002426305605526, −2.80485342721307959585270481488, −1.56935351233067698973222335019, 1.76830034427994705450478633633, 3.13432076860361934394770720077, 3.92227376923421779888313224465, 5.27528263382430926737567319471, 6.68428342753002852130323942353, 7.20668869624099408752714245691, 8.378590181040095552433545087593, 9.188809932071717020813003294111, 10.32296642448494467941686976052, 10.67588452810729918791856404390

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