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Properties

Label 2-507-39.5-c1-0-41
Degree $2$
Conductor $507$
Sign $-0.998 - 0.0605i$
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  + (−0.249 − 0.249i)2-s + (0.892 − 1.48i)3-s − 1.87i·4-s + (−2.45 − 2.45i)5-s + (−0.592 + 0.147i)6-s + (0.821 + 0.821i)7-s + (−0.965 + 0.965i)8-s + (−1.40 − 2.64i)9-s + 1.22i·10-s + (−1.32 + 1.32i)11-s + (−2.78 − 1.67i)12-s − 0.409i·14-s + (−5.84 + 1.45i)15-s − 3.27·16-s + 5.90·17-s + (−0.309 + 1.01i)18-s + ⋯
L(s)  = 1  + (−0.176 − 0.176i)2-s + (0.515 − 0.857i)3-s − 0.937i·4-s + (−1.09 − 1.09i)5-s + (−0.241 + 0.0602i)6-s + (0.310 + 0.310i)7-s + (−0.341 + 0.341i)8-s + (−0.469 − 0.882i)9-s + 0.387i·10-s + (−0.399 + 0.399i)11-s + (−0.803 − 0.483i)12-s − 0.109i·14-s + (−1.50 + 0.375i)15-s − 0.817·16-s + 1.43·17-s + (−0.0728 + 0.238i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.998 - 0.0605i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.998 - 0.0605i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0315751 + 1.04152i\)
\(L(\frac12)\) \(\approx\) \(0.0315751 + 1.04152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.892 + 1.48i)T \)
13 \( 1 \)
good2 \( 1 + (0.249 + 0.249i)T + 2iT^{2} \)
5 \( 1 + (2.45 + 2.45i)T + 5iT^{2} \)
7 \( 1 + (-0.821 - 0.821i)T + 7iT^{2} \)
11 \( 1 + (1.32 - 1.32i)T - 11iT^{2} \)
17 \( 1 - 5.90T + 17T^{2} \)
19 \( 1 + (-3.48 + 3.48i)T - 19iT^{2} \)
23 \( 1 + 2.70T + 23T^{2} \)
29 \( 1 - 2.68iT - 29T^{2} \)
31 \( 1 + (3.22 - 3.22i)T - 31iT^{2} \)
37 \( 1 + (-1.52 - 1.52i)T + 37iT^{2} \)
41 \( 1 + (4.81 + 4.81i)T + 41iT^{2} \)
43 \( 1 + 5.55iT - 43T^{2} \)
47 \( 1 + (-2.23 + 2.23i)T - 47iT^{2} \)
53 \( 1 - 2.46iT - 53T^{2} \)
59 \( 1 + (-7.07 + 7.07i)T - 59iT^{2} \)
61 \( 1 + 2.66T + 61T^{2} \)
67 \( 1 + (-4.81 + 4.81i)T - 67iT^{2} \)
71 \( 1 + (8.20 + 8.20i)T + 71iT^{2} \)
73 \( 1 + (9.13 + 9.13i)T + 73iT^{2} \)
79 \( 1 - 1.10T + 79T^{2} \)
83 \( 1 + (-4.58 - 4.58i)T + 83iT^{2} \)
89 \( 1 + (3.02 - 3.02i)T - 89iT^{2} \)
97 \( 1 + (-8.67 + 8.67i)T - 97iT^{2} \)
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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

−10.41740494134572854585396072123, −9.338822713169517773264044368944, −8.664771089161430656092552147939, −7.85219796187827275561925814448, −7.05536772015937683117211539072, −5.60892737117165328467801774670, −4.90050343977764131446304095658, −3.40641716426120741382906158705, −1.82795561071198707958780819892, −0.63096704366140229064362185975, 2.84957369604267366698200550474, 3.52377794262439907923155144791, 4.27348846772650047242888900097, 5.79469663542163566611316796809, 7.34709810869113048408974718155, 7.79405574057148937280725270559, 8.369010393836481139734669851695, 9.678070175329536633074478630286, 10.42156205040392597093668503416, 11.41530951157649601187543402250

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