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Properties

Label 2-2e2-4.3-c34-0-10
Degree $2$
Conductor $4$
Sign $0.978 + 0.205i$
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  + (1.35e4 − 1.30e5i)2-s + 2.10e8i·3-s + (−1.68e10 − 3.53e9i)4-s + 1.26e12·5-s + (2.74e13 + 2.85e12i)6-s − 2.66e14i·7-s + (−6.87e14 + 2.14e15i)8-s − 2.78e16·9-s + (1.70e16 − 1.64e17i)10-s − 3.69e16i·11-s + (7.44e17 − 3.54e18i)12-s + 5.25e18·13-s + (−3.48e19 − 3.61e18i)14-s + 2.66e20i·15-s + (2.70e20 + 1.18e20i)16-s + 7.43e20·17-s + ⋯
L(s)  = 1  + (0.103 − 0.994i)2-s + 1.63i·3-s + (−0.978 − 0.205i)4-s + 1.65·5-s + (1.62 + 0.168i)6-s − 1.14i·7-s + (−0.305 + 0.952i)8-s − 1.66·9-s + (0.170 − 1.64i)10-s − 0.0730i·11-s + (0.335 − 1.59i)12-s + 0.607·13-s + (−1.14 − 0.118i)14-s + 2.70i·15-s + (0.915 + 0.402i)16-s + 0.898·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.978 + 0.205i$
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: \((2,\ 4,\ (\ :17),\ 0.978 + 0.205i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(2.623127347\)
\(L(\frac12)\) \(\approx\) \(2.623127347\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35e4 + 1.30e5i)T \)
good3 \( 1 - 2.10e8iT - 1.66e16T^{2} \)
5 \( 1 - 1.26e12T + 5.82e23T^{2} \)
7 \( 1 + 2.66e14iT - 5.41e28T^{2} \)
11 \( 1 + 3.69e16iT - 2.55e35T^{2} \)
13 \( 1 - 5.25e18T + 7.48e37T^{2} \)
17 \( 1 - 7.43e20T + 6.84e41T^{2} \)
19 \( 1 + 5.11e21iT - 3.00e43T^{2} \)
23 \( 1 - 3.98e22iT - 1.98e46T^{2} \)
29 \( 1 + 1.21e23T + 5.26e49T^{2} \)
31 \( 1 - 3.15e25iT - 5.08e50T^{2} \)
37 \( 1 - 8.45e26T + 2.08e53T^{2} \)
41 \( 1 - 3.33e27T + 6.83e54T^{2} \)
43 \( 1 + 1.26e26iT - 3.45e55T^{2} \)
47 \( 1 + 6.15e27iT - 7.10e56T^{2} \)
53 \( 1 - 1.72e29T + 4.22e58T^{2} \)
59 \( 1 - 4.94e29iT - 1.61e60T^{2} \)
61 \( 1 + 3.28e30T + 5.02e60T^{2} \)
67 \( 1 - 1.65e31iT - 1.22e62T^{2} \)
71 \( 1 + 5.10e30iT - 8.76e62T^{2} \)
73 \( 1 - 3.84e29T + 2.25e63T^{2} \)
79 \( 1 + 9.83e31iT - 3.30e64T^{2} \)
83 \( 1 + 2.13e32iT - 1.77e65T^{2} \)
89 \( 1 + 6.24e32T + 1.90e66T^{2} \)
97 \( 1 - 4.77e33T + 3.55e67T^{2} \)
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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

−16.63482677507109306392310990478, −14.43443625850361348108823919793, −13.41376703723359621120340913528, −10.87767620347889533736489925961, −10.08606750534884164660503860414, −9.147518592868290273636039823964, −5.61487419053258455923603699085, −4.33889407456650192835417730570, −2.95405195612938035155144411084, −1.09313244416236281959965221241, 1.10979193040354849194613693049, 2.41669861344605221151815117843, 5.76541855763290231283193000199, 6.17459356160489368709795711040, 7.933632400229419391781974050214, 9.388990103220368739435869434904, 12.44136971265996916922895662849, 13.42151169524142758836541227595, 14.54052711450277974058538866413, 16.86237078031218385760438862987

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