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Properties

Label 2-507-13.12-c1-0-9
Degree $2$
Conductor $507$
Sign $0.832 + 0.554i$
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  − i·2-s − 3-s + 4-s + i·5-s + i·6-s + 2i·7-s − 3i·8-s + 9-s + 10-s − 2i·11-s − 12-s + 2·14-s − i·15-s − 16-s + 7·17-s − i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s + 0.5·4-s + 0.447i·5-s + 0.408i·6-s + 0.755i·7-s − 1.06i·8-s + 0.333·9-s + 0.316·10-s − 0.603i·11-s − 0.288·12-s + 0.534·14-s − 0.258i·15-s − 0.250·16-s + 1.69·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42341 - 0.430976i\)
\(L(\frac12)\) \(\approx\) \(1.42341 - 0.430976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + iT - 2T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - iT - 37T^{2} \)
41 \( 1 + 9iT - 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 - 11iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−10.88302241893883361078222358039, −10.24001098122455791523737200437, −9.368930806078042498823291254679, −8.080677528807436121356495576453, −7.09045611129576374140945211260, −6.08812833965906048320506583663, −5.37490673361024309327036264971, −3.67243749731344640759025038789, −2.79844164800277790581293115704, −1.31025962131053228163622230969, 1.24294828782970952351583060893, 3.08186222949977246792553401651, 4.74862985842187966766621402708, 5.32945675881602155587182282086, 6.61560568041594322329701925057, 7.16565704900569498187389351720, 8.010402834333492000945264505915, 9.162590430923933221910568357790, 10.23954221115180697276067613211, 10.98472859334651605966597579960

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