↑

Properties

Label 2-2e2-4.3-c34-0-1
Degree $2$
Conductor $4$
Sign $0.977 - 0.211i$
Analytic cond. $29.2902$
Root an. cond. $5.41204$
Motivic weight $34$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39e4 − 1.30e5i)2-s − 2.06e8i·3-s + (−1.67e10 + 3.62e9i)4-s − 9.83e10·5-s + (−2.69e13 + 2.87e12i)6-s − 2.44e14i·7-s + (7.06e14 + 2.13e15i)8-s − 2.59e16·9-s + (1.36e15 + 1.28e16i)10-s + 8.27e17i·11-s + (7.48e17 + 3.46e18i)12-s − 1.42e15·13-s + (−3.18e19 + 3.40e18i)14-s + 2.03e19i·15-s + (2.68e20 − 1.21e20i)16-s − 8.57e20·17-s + ⋯
L(s)  = 1  + (−0.106 − 0.994i)2-s − 1.59i·3-s + (−0.977 + 0.211i)4-s − 0.128·5-s + (−1.58 + 0.169i)6-s − 1.05i·7-s + (0.313 + 0.949i)8-s − 1.55·9-s + (0.0136 + 0.128i)10-s + 1.63i·11-s + (0.337 + 1.56i)12-s − 0.000165·13-s + (−1.04 + 0.111i)14-s + 0.206i·15-s + (0.910 − 0.412i)16-s − 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.977 - 0.211i$
Analytic conductor: \(29.2902\)
Root analytic conductor: \(5.41204\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :17),\ 0.977 - 0.211i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.1235719252\)
\(L(\frac12)\) \(\approx\) \(0.1235719252\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39e4 + 1.30e5i)T \)
good3 \( 1 + 2.06e8iT - 1.66e16T^{2} \)
5 \( 1 + 9.83e10T + 5.82e23T^{2} \)
7 \( 1 + 2.44e14iT - 5.41e28T^{2} \)
11 \( 1 - 8.27e17iT - 2.55e35T^{2} \)
13 \( 1 + 1.42e15T + 7.48e37T^{2} \)
17 \( 1 + 8.57e20T + 6.84e41T^{2} \)
19 \( 1 - 6.28e21iT - 3.00e43T^{2} \)
23 \( 1 + 2.11e23iT - 1.98e46T^{2} \)
29 \( 1 - 1.04e25T + 5.26e49T^{2} \)
31 \( 1 - 4.38e24iT - 5.08e50T^{2} \)
37 \( 1 + 1.02e26T + 2.08e53T^{2} \)
41 \( 1 + 1.40e27T + 6.83e54T^{2} \)
43 \( 1 - 3.94e27iT - 3.45e55T^{2} \)
47 \( 1 + 3.15e28iT - 7.10e56T^{2} \)
53 \( 1 + 3.97e29T + 4.22e58T^{2} \)
59 \( 1 - 1.46e30iT - 1.61e60T^{2} \)
61 \( 1 - 1.24e30T + 5.02e60T^{2} \)
67 \( 1 - 1.03e31iT - 1.22e62T^{2} \)
71 \( 1 - 1.84e31iT - 8.76e62T^{2} \)
73 \( 1 + 2.03e30T + 2.25e63T^{2} \)
79 \( 1 - 4.21e31iT - 3.30e64T^{2} \)
83 \( 1 - 3.72e32iT - 1.77e65T^{2} \)
89 \( 1 + 1.82e33T + 1.90e66T^{2} \)
97 \( 1 - 6.65e33T + 3.55e67T^{2} \)
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

−17.31730674099033189258932416422, −14.17995069972204314492353807903, −12.95968668638560856372638991438, −12.00936074819788881483785643804, −10.24100602048082138298168355979, −8.125654098007581854941788534566, −6.84298901415896491256886190028, −4.33364146323457287341709485333, −2.29979147870366534863031816351, −1.26897052935649912368545642270, 0.04370443037541181685944836283, 3.32145500714898416308484819926, 4.83175128189809705672455478511, 5.99374063003472313448173734213, 8.514013331015905305807691612416, 9.421036441679485181943638024923, 11.18445399892478412997814535019, 13.81330817385170475206136624428, 15.51443595463379231534430463189, 15.83615065183545763930447455292

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