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Properties

Label 2-2e2-4.3-c34-0-0
Degree $2$
Conductor $4$
Sign $-0.846 + 0.532i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.25e5 − 3.63e4i)2-s + 1.93e8i·3-s + (1.45e10 − 9.14e9i)4-s − 8.66e11·5-s + (7.02e12 + 2.43e13i)6-s − 8.12e13i·7-s + (1.49e15 − 1.68e15i)8-s − 2.07e16·9-s + (−1.09e17 + 3.14e16i)10-s + 5.42e17i·11-s + (1.76e18 + 2.81e18i)12-s − 1.50e19·13-s + (−2.95e18 − 1.02e19i)14-s − 1.67e20i·15-s + (1.27e20 − 2.66e20i)16-s − 1.97e20·17-s + ⋯
L(s)  = 1  + (0.960 − 0.277i)2-s + 1.49i·3-s + (0.846 − 0.532i)4-s − 1.13·5-s + (0.415 + 1.43i)6-s − 0.349i·7-s + (0.665 − 0.746i)8-s − 1.24·9-s + (−1.09 + 0.314i)10-s + 1.07i·11-s + (0.797 + 1.26i)12-s − 1.73·13-s + (−0.0967 − 0.335i)14-s − 1.70i·15-s + (0.432 − 0.901i)16-s − 0.238·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.846 + 0.532i$
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: \((2,\ 4,\ (\ :17),\ -0.846 + 0.532i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.1487307861\)
\(L(\frac12)\) \(\approx\) \(0.1487307861\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25e5 + 3.63e4i)T \)
good3 \( 1 - 1.93e8iT - 1.66e16T^{2} \)
5 \( 1 + 8.66e11T + 5.82e23T^{2} \)
7 \( 1 + 8.12e13iT - 5.41e28T^{2} \)
11 \( 1 - 5.42e17iT - 2.55e35T^{2} \)
13 \( 1 + 1.50e19T + 7.48e37T^{2} \)
17 \( 1 + 1.97e20T + 6.84e41T^{2} \)
19 \( 1 + 6.27e21iT - 3.00e43T^{2} \)
23 \( 1 + 2.46e23iT - 1.98e46T^{2} \)
29 \( 1 + 7.75e24T + 5.26e49T^{2} \)
31 \( 1 - 1.56e24iT - 5.08e50T^{2} \)
37 \( 1 - 9.31e25T + 2.08e53T^{2} \)
41 \( 1 + 2.42e27T + 6.83e54T^{2} \)
43 \( 1 - 8.07e27iT - 3.45e55T^{2} \)
47 \( 1 - 3.66e28iT - 7.10e56T^{2} \)
53 \( 1 + 9.36e28T + 4.22e58T^{2} \)
59 \( 1 - 1.58e30iT - 1.61e60T^{2} \)
61 \( 1 + 2.50e30T + 5.02e60T^{2} \)
67 \( 1 + 3.53e30iT - 1.22e62T^{2} \)
71 \( 1 - 2.74e31iT - 8.76e62T^{2} \)
73 \( 1 - 2.97e31T + 2.25e63T^{2} \)
79 \( 1 + 3.89e31iT - 3.30e64T^{2} \)
83 \( 1 - 1.42e32iT - 1.77e65T^{2} \)
89 \( 1 + 9.61e32T + 1.90e66T^{2} \)
97 \( 1 + 1.00e34T + 3.55e67T^{2} \)
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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

−16.70182857145942706598745334130, −15.37214905737528943417837684394, −14.73187299231382567989644631831, −12.39637699061635443387258546461, −11.00505031545854542133313948482, −9.737516424357559195142094012688, −7.24247054190421723432495175188, −4.74299910577326723588855690389, −4.29743277219232242394727759639, −2.72433726568946652421558102877, 0.03101637715391467973773578133, 1.95397172089531446702733261190, 3.50217035057573867991593486181, 5.56795818409594452814946503448, 7.17826740121118541165522421261, 7.999048758861229607764439363594, 11.61250334733851020074686489928, 12.33434931883553527548684578691, 13.73946937311523901065578980841, 15.22859035621667448084012264306

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