Properties

Label 2-5-5.4-c33-0-8
Degree $2$
Conductor $5$
Sign $0.847 - 0.530i$
Analytic cond. $34.4914$
Root an. cond. $5.87293$
Motivic weight $33$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.77e4i·2-s + 6.45e7i·3-s + 7.82e9·4-s + (−2.89e11 + 1.80e11i)5-s + 1.78e12·6-s − 3.09e13i·7-s − 4.55e14i·8-s + 1.39e15·9-s + (5.01e15 + 8.01e15i)10-s + 1.96e17·11-s + 5.04e17i·12-s − 4.53e18i·13-s − 8.57e17·14-s + (−1.16e19 − 1.86e19i)15-s + 5.45e19·16-s + 2.94e20i·17-s + ⋯
L(s)  = 1  − 0.299i·2-s + 0.865i·3-s + 0.910·4-s + (−0.847 + 0.530i)5-s + 0.258·6-s − 0.351i·7-s − 0.571i·8-s + 0.251·9-s + (0.158 + 0.253i)10-s + 1.28·11-s + 0.787i·12-s − 1.88i·13-s − 0.105·14-s + (−0.458 − 0.733i)15-s + 0.739·16-s + 1.46i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.847 - 0.530i$
Analytic conductor: \(34.4914\)
Root analytic conductor: \(5.87293\)
Motivic weight: \(33\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :33/2),\ 0.847 - 0.530i)\)

Particular Values

\(L(17)\) \(\approx\) \(2.525797979\)
\(L(\frac12)\) \(\approx\) \(2.525797979\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.89e11 - 1.80e11i)T \)
good2 \( 1 + 2.77e4iT - 8.58e9T^{2} \)
3 \( 1 - 6.45e7iT - 5.55e15T^{2} \)
7 \( 1 + 3.09e13iT - 7.73e27T^{2} \)
11 \( 1 - 1.96e17T + 2.32e34T^{2} \)
13 \( 1 + 4.53e18iT - 5.75e36T^{2} \)
17 \( 1 - 2.94e20iT - 4.02e40T^{2} \)
19 \( 1 + 6.15e20T + 1.58e42T^{2} \)
23 \( 1 - 1.65e22iT - 8.65e44T^{2} \)
29 \( 1 - 9.17e22T + 1.81e48T^{2} \)
31 \( 1 - 5.75e24T + 1.64e49T^{2} \)
37 \( 1 - 9.32e25iT - 5.63e51T^{2} \)
41 \( 1 - 4.59e26T + 1.66e53T^{2} \)
43 \( 1 - 2.98e26iT - 8.02e53T^{2} \)
47 \( 1 - 3.83e27iT - 1.51e55T^{2} \)
53 \( 1 + 1.34e28iT - 7.96e56T^{2} \)
59 \( 1 + 1.39e29T + 2.74e58T^{2} \)
61 \( 1 + 1.27e29T + 8.23e58T^{2} \)
67 \( 1 + 1.55e30iT - 1.82e60T^{2} \)
71 \( 1 - 4.80e30T + 1.23e61T^{2} \)
73 \( 1 - 2.54e30iT - 3.08e61T^{2} \)
79 \( 1 + 1.90e31T + 4.18e62T^{2} \)
83 \( 1 - 1.69e31iT - 2.13e63T^{2} \)
89 \( 1 - 8.93e31T + 2.13e64T^{2} \)
97 \( 1 - 6.99e32iT - 3.65e65T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.59044174624463283934122871130, −14.96915391222112998203337347452, −12.46876022711655236984423784336, −10.98676575311603066251101367964, −10.15126609951797230971997197440, −7.900064662779545501849474259975, −6.36856823467558834474314239060, −4.09978249318805670901826062251, −3.14725876519792720652581923406, −1.11205112994563700428511931981, 0.962016027512592593022638201663, 2.19417139513139080618556006359, 4.30393660887613456647790715850, 6.50287378062508941756587404473, 7.33162527604516630437690261749, 8.970552929463708667807567272157, 11.59107087599806358292061639173, 12.17377331837217645847362272287, 14.18380884334344382236437586608, 15.81630247477563510575180421466

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