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Properties

Label 2-507-507.332-c1-0-13
Degree $2$
Conductor $507$
Sign $-0.734 - 0.678i$
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  + (0.742 + 1.56i)3-s + (−0.633 + 1.89i)4-s + (−0.0345 − 0.0318i)7-s + (−1.89 + 2.32i)9-s + (−3.43 + 0.417i)12-s + (2.59 + 2.5i)13-s + (−3.19 − 2.40i)16-s + (−1.66 + 6.22i)19-s + (0.0242 − 0.0776i)21-s + (−1.19 − 4.85i)25-s + (−5.04 − 1.24i)27-s + (0.0823 − 0.0453i)28-s + (−3.34 − 5.53i)31-s + (−3.20 − 5.07i)36-s + (0.233 + 11.6i)37-s + ⋯
L(s)  = 1  + (0.428 + 0.903i)3-s + (−0.316 + 0.948i)4-s + (−0.0130 − 0.0120i)7-s + (−0.632 + 0.774i)9-s + (−0.992 + 0.120i)12-s + (0.720 + 0.693i)13-s + (−0.799 − 0.600i)16-s + (−0.382 + 1.42i)19-s + (0.00528 − 0.0169i)21-s + (−0.239 − 0.970i)25-s + (−0.970 − 0.239i)27-s + (0.0155 − 0.00856i)28-s + (−0.600 − 0.993i)31-s + (−0.534 − 0.845i)36-s + (0.0384 + 1.90i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.734 - 0.678i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.484877 + 1.24022i\)
\(L(\frac12)\) \(\approx\) \(0.484877 + 1.24022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.742 - 1.56i)T \)
13 \( 1 + (-2.59 - 2.5i)T \)
good2 \( 1 + (0.633 - 1.89i)T^{2} \)
5 \( 1 + (1.19 + 4.85i)T^{2} \)
7 \( 1 + (0.0345 + 0.0318i)T + (0.563 + 6.97i)T^{2} \)
11 \( 1 + (-10.7 - 2.20i)T^{2} \)
17 \( 1 + (-16.9 + 1.36i)T^{2} \)
19 \( 1 + (1.66 - 6.22i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-27.5 - 9.18i)T^{2} \)
31 \( 1 + (3.34 + 5.53i)T + (-14.4 + 27.4i)T^{2} \)
37 \( 1 + (-0.233 - 11.6i)T + (-36.9 + 1.48i)T^{2} \)
41 \( 1 + (31.7 + 25.9i)T^{2} \)
43 \( 1 + (-5.46 + 5.24i)T + (1.73 - 42.9i)T^{2} \)
47 \( 1 + (-46.6 + 5.66i)T^{2} \)
53 \( 1 + (-30.1 - 43.6i)T^{2} \)
59 \( 1 + (56.6 + 16.4i)T^{2} \)
61 \( 1 + (-12.7 - 8.07i)T + (26.1 + 55.1i)T^{2} \)
67 \( 1 + (-14.0 - 7.00i)T + (40.2 + 53.5i)T^{2} \)
71 \( 1 + (11.3 - 70.0i)T^{2} \)
73 \( 1 + (-4.86 - 10.8i)T + (-48.4 + 54.6i)T^{2} \)
79 \( 1 + (-13.1 - 11.6i)T + (9.52 + 78.4i)T^{2} \)
83 \( 1 + (77.6 + 29.4i)T^{2} \)
89 \( 1 + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.21 - 3.70i)T + (69.9 + 67.1i)T^{2} \)
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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

−11.27710105337593609295399113575, −10.19747145606093973361279357936, −9.463808626426432791571432187043, −8.419974998951910434221700997281, −8.113119205879334059456643462898, −6.74501324785539150870960282704, −5.47874372342379620884910595588, −4.17004727453865055295159009950, −3.73146270332195913559891312787, −2.35567898762430099576232644281, 0.77238524613489413908614373521, 2.14780600133109017757787238483, 3.56503038371579162115933452069, 5.06951973307305358002022319159, 5.99875145905097466392133413543, 6.85744360875322717457168299626, 7.85807940391954552594762395664, 8.947549321907720673855387255503, 9.386681853446896689212972920380, 10.79012546688153410603335368550

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