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Properties

Label 2-2e2-4.3-c34-0-8
Degree $2$
Conductor $4$
Sign $-0.525 - 0.850i$
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  + (1.14e5 + 6.38e4i)2-s + 8.29e7i·3-s + (9.03e9 + 1.46e10i)4-s + 6.47e11·5-s + (−5.29e12 + 9.49e12i)6-s + 1.29e14i·7-s + (1.01e14 + 2.24e15i)8-s + 9.79e15·9-s + (7.41e16 + 4.13e16i)10-s − 3.33e16i·11-s + (−1.21e18 + 7.49e17i)12-s + 8.70e18·13-s + (−8.27e18 + 1.48e19i)14-s + 5.37e19i·15-s + (−1.31e20 + 2.64e20i)16-s − 4.22e20·17-s + ⋯
L(s)  = 1  + (0.873 + 0.486i)2-s + 0.642i·3-s + (0.525 + 0.850i)4-s + 0.848·5-s + (−0.312 + 0.561i)6-s + 0.557i·7-s + (0.0450 + 0.998i)8-s + 0.587·9-s + (0.741 + 0.413i)10-s − 0.0659i·11-s + (−0.546 + 0.337i)12-s + 1.00·13-s + (−0.271 + 0.486i)14-s + 0.545i·15-s + (−0.447 + 0.894i)16-s − 0.511·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.525 - 0.850i$
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: \((2,\ 4,\ (\ :17),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(4.124344613\)
\(L(\frac12)\) \(\approx\) \(4.124344613\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14e5 - 6.38e4i)T \)
good3 \( 1 - 8.29e7iT - 1.66e16T^{2} \)
5 \( 1 - 6.47e11T + 5.82e23T^{2} \)
7 \( 1 - 1.29e14iT - 5.41e28T^{2} \)
11 \( 1 + 3.33e16iT - 2.55e35T^{2} \)
13 \( 1 - 8.70e18T + 7.48e37T^{2} \)
17 \( 1 + 4.22e20T + 6.84e41T^{2} \)
19 \( 1 - 4.20e21iT - 3.00e43T^{2} \)
23 \( 1 + 2.57e23iT - 1.98e46T^{2} \)
29 \( 1 + 4.39e24T + 5.26e49T^{2} \)
31 \( 1 - 3.47e25iT - 5.08e50T^{2} \)
37 \( 1 + 5.87e26T + 2.08e53T^{2} \)
41 \( 1 - 1.01e27T + 6.83e54T^{2} \)
43 \( 1 + 2.83e27iT - 3.45e55T^{2} \)
47 \( 1 + 2.29e28iT - 7.10e56T^{2} \)
53 \( 1 - 1.06e29T + 4.22e58T^{2} \)
59 \( 1 + 4.19e28iT - 1.61e60T^{2} \)
61 \( 1 - 3.59e30T + 5.02e60T^{2} \)
67 \( 1 + 1.73e31iT - 1.22e62T^{2} \)
71 \( 1 + 4.95e31iT - 8.76e62T^{2} \)
73 \( 1 - 8.07e31T + 2.25e63T^{2} \)
79 \( 1 - 2.83e32iT - 3.30e64T^{2} \)
83 \( 1 + 2.69e32iT - 1.77e65T^{2} \)
89 \( 1 + 3.78e32T + 1.90e66T^{2} \)
97 \( 1 - 1.06e33T + 3.55e67T^{2} \)
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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

−16.51818596802183127282689841315, −15.38381305695240348417415555912, −13.93345284712240541567787695264, −12.52541311354837448752775544506, −10.57299069120386077867528560462, −8.712595225867707104432661689201, −6.54479406147488898905668494093, −5.23636957401969271167204487543, −3.74777916558673262201068309928, −2.00228834143390156668787101981, 1.03266288384757778830726289236, 2.08321204623520256439199838210, 3.94920700264006437764127818152, 5.76010454520505011699420159437, 7.10801148505267237076473136185, 9.727519352115691415844192187756, 11.28204243666451480955888881214, 13.13635093376174404365200481188, 13.67509313044873996969169850514, 15.60646571535982884360391335891

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