Properties

Label 2-2e2-4.3-c34-0-2
Degree $2$
Conductor $4$
Sign $0.182 - 0.983i$
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  + (−8.37e4 − 1.00e5i)2-s + 9.86e7i·3-s + (−3.13e9 + 1.68e10i)4-s − 1.17e12·5-s + (9.94e12 − 8.26e12i)6-s − 2.80e14i·7-s + (1.96e15 − 1.09e15i)8-s + 6.93e15·9-s + (9.82e16 + 1.18e17i)10-s − 6.38e17i·11-s + (−1.66e18 − 3.09e17i)12-s + 9.36e18·13-s + (−2.82e19 + 2.35e19i)14-s − 1.15e20i·15-s + (−2.75e20 − 1.06e20i)16-s − 1.12e21·17-s + ⋯
L(s)  = 1  + (−0.639 − 0.769i)2-s + 0.764i·3-s + (−0.182 + 0.983i)4-s − 1.53·5-s + (0.587 − 0.488i)6-s − 1.20i·7-s + (0.872 − 0.487i)8-s + 0.415·9-s + (0.982 + 1.18i)10-s − 1.26i·11-s + (−0.751 − 0.139i)12-s + 1.08·13-s + (−0.928 + 0.771i)14-s − 1.17i·15-s + (−0.933 − 0.359i)16-s − 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.182 - 0.983i$
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: \((2,\ 4,\ (\ :17),\ 0.182 - 0.983i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.4349682099\)
\(L(\frac12)\) \(\approx\) \(0.4349682099\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8.37e4 + 1.00e5i)T \)
good3 \( 1 - 9.86e7iT - 1.66e16T^{2} \)
5 \( 1 + 1.17e12T + 5.82e23T^{2} \)
7 \( 1 + 2.80e14iT - 5.41e28T^{2} \)
11 \( 1 + 6.38e17iT - 2.55e35T^{2} \)
13 \( 1 - 9.36e18T + 7.48e37T^{2} \)
17 \( 1 + 1.12e21T + 6.84e41T^{2} \)
19 \( 1 - 4.37e21iT - 3.00e43T^{2} \)
23 \( 1 - 7.31e22iT - 1.98e46T^{2} \)
29 \( 1 + 1.01e25T + 5.26e49T^{2} \)
31 \( 1 + 4.87e23iT - 5.08e50T^{2} \)
37 \( 1 + 6.91e25T + 2.08e53T^{2} \)
41 \( 1 - 1.16e27T + 6.83e54T^{2} \)
43 \( 1 + 5.88e27iT - 3.45e55T^{2} \)
47 \( 1 + 1.68e27iT - 7.10e56T^{2} \)
53 \( 1 - 3.41e28T + 4.22e58T^{2} \)
59 \( 1 - 2.08e30iT - 1.61e60T^{2} \)
61 \( 1 + 5.31e29T + 5.02e60T^{2} \)
67 \( 1 - 2.05e31iT - 1.22e62T^{2} \)
71 \( 1 + 8.65e30iT - 8.76e62T^{2} \)
73 \( 1 + 2.42e30T + 2.25e63T^{2} \)
79 \( 1 - 9.47e31iT - 3.30e64T^{2} \)
83 \( 1 - 4.82e32iT - 1.77e65T^{2} \)
89 \( 1 + 8.20e32T + 1.90e66T^{2} \)
97 \( 1 - 3.81e33T + 3.55e67T^{2} \)
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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

−16.61707466903713784691070587230, −15.74321012703704537648959690788, −13.33853625032165337340742598033, −11.35169782040042204845722284479, −10.62266775203247263501359475967, −8.721644530239821150569398269861, −7.38776829355042442878486266811, −4.01464961390550048771809279626, −3.69061105383856926704727294813, −0.995784800873868343617475605959, 0.22166409380604966581265957004, 1.92004680346816415903108190133, 4.50839124202331904840598960315, 6.57976868325170658222254822861, 7.71918297224173017395556752718, 8.978715827937292788395675184940, 11.30029468977057771440834665993, 12.82812880672242587814606673423, 15.19836034902222045398354982663, 15.77395560227361001393369949854

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