Properties

Degree $2$
Conductor $4$
Sign $-0.186 + 0.982i$
Motivic weight $36$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67e5 − 2.01e5i)2-s − 6.14e8i·3-s + (−1.28e10 + 6.75e10i)4-s − 3.33e12·5-s + (−1.24e14 + 1.02e14i)6-s + 2.74e15i·7-s + (1.57e16 − 8.70e15i)8-s − 2.27e17·9-s + (5.57e17 + 6.73e17i)10-s − 5.16e18i·11-s + (4.14e19 + 7.86e18i)12-s + 1.38e20·13-s + (5.53e20 − 4.58e20i)14-s + 2.04e21i·15-s + (−4.39e21 − 1.72e21i)16-s − 2.51e21·17-s + ⋯
L(s)  = 1  + (−0.637 − 0.770i)2-s − 1.58i·3-s + (−0.186 + 0.982i)4-s − 0.873·5-s + (−1.22 + 1.01i)6-s + 1.68i·7-s + (0.875 − 0.483i)8-s − 1.51·9-s + (0.557 + 0.673i)10-s − 0.929i·11-s + (1.55 + 0.295i)12-s + 1.23·13-s + (1.29 − 1.07i)14-s + 1.38i·15-s + (−0.930 − 0.366i)16-s − 0.179·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.186 + 0.982i$
Motivic weight: \(36\)
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :18),\ -0.186 + 0.982i)\)

Particular Values

\(L(\frac{37}{2})\) \(\approx\) \(1.044964863\)
\(L(\frac12)\) \(\approx\) \(1.044964863\)
\(L(19)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.67e5 + 2.01e5i)T \)
good3 \( 1 + 6.14e8iT - 1.50e17T^{2} \)
5 \( 1 + 3.33e12T + 1.45e25T^{2} \)
7 \( 1 - 2.74e15iT - 2.65e30T^{2} \)
11 \( 1 + 5.16e18iT - 3.09e37T^{2} \)
13 \( 1 - 1.38e20T + 1.26e40T^{2} \)
17 \( 1 + 2.51e21T + 1.97e44T^{2} \)
19 \( 1 - 5.92e22iT - 1.08e46T^{2} \)
23 \( 1 - 3.10e24iT - 1.05e49T^{2} \)
29 \( 1 - 3.37e26T + 4.42e52T^{2} \)
31 \( 1 + 6.60e26iT - 4.88e53T^{2} \)
37 \( 1 + 1.26e28T + 2.85e56T^{2} \)
41 \( 1 - 3.78e28T + 1.14e58T^{2} \)
43 \( 1 - 2.97e29iT - 6.38e58T^{2} \)
47 \( 1 + 8.86e29iT - 1.56e60T^{2} \)
53 \( 1 - 8.62e30T + 1.18e62T^{2} \)
59 \( 1 - 1.68e31iT - 5.63e63T^{2} \)
61 \( 1 - 7.55e31T + 1.87e64T^{2} \)
67 \( 1 + 5.97e32iT - 5.47e65T^{2} \)
71 \( 1 + 1.32e33iT - 4.41e66T^{2} \)
73 \( 1 - 2.88e33T + 1.20e67T^{2} \)
79 \( 1 + 1.29e34iT - 2.06e68T^{2} \)
83 \( 1 + 3.72e34iT - 1.22e69T^{2} \)
89 \( 1 - 1.48e35T + 1.50e70T^{2} \)
97 \( 1 - 8.33e35T + 3.34e71T^{2} \)
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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

−15.88507406842921333162090419814, −13.44206229989723350333785967285, −12.12979992408297750868871398331, −11.45224732623590628386328875214, −8.710001256801155623848966814408, −7.947145951889296932569035357209, −6.12239334690366963878336051818, −3.26595773250465964424027437433, −1.94451972714920004295378240031, −0.68385309521737848678694755155, 0.72137168172718499739154869840, 3.88983626827509775119334485524, 4.69102613237241381783991972126, 6.94083898032376367484638670597, 8.516294371487449216900220487013, 10.12943795318618998207601582378, 10.90427043210579713834680956318, 14.06117234821898622734710642634, 15.50222679794837000309241894494, 16.24042216346788348399667859908

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