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Properties

Label 2-1045-5.4-c1-0-0
Degree $2$
Conductor $1045$
Sign $-0.983 + 0.183i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.84i·2-s + 3.33i·3-s − 1.40·4-s + (−2.19 + 0.409i)5-s + 6.14·6-s + 3.04i·7-s − 1.10i·8-s − 8.09·9-s + (0.756 + 4.05i)10-s − 11-s − 4.67i·12-s + 2.03i·13-s + 5.62·14-s + (−1.36 − 7.32i)15-s − 4.83·16-s − 6.84i·17-s + ⋯
L(s)  = 1  − 1.30i·2-s + 1.92i·3-s − 0.701·4-s + (−0.983 + 0.183i)5-s + 2.50·6-s + 1.15i·7-s − 0.389i·8-s − 2.69·9-s + (0.239 + 1.28i)10-s − 0.301·11-s − 1.34i·12-s + 0.563i·13-s + 1.50·14-s + (−0.352 − 1.89i)15-s − 1.20·16-s − 1.66i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.983 + 0.183i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -0.983 + 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1236152342\)
\(L(\frac12)\) \(\approx\) \(0.1236152342\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.19 - 0.409i)T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 + 1.84iT - 2T^{2} \)
3 \( 1 - 3.33iT - 3T^{2} \)
7 \( 1 - 3.04iT - 7T^{2} \)
13 \( 1 - 2.03iT - 13T^{2} \)
17 \( 1 + 6.84iT - 17T^{2} \)
23 \( 1 - 5.01iT - 23T^{2} \)
29 \( 1 + 7.08T + 29T^{2} \)
31 \( 1 - 5.61T + 31T^{2} \)
37 \( 1 + 6.54iT - 37T^{2} \)
41 \( 1 + 8.63T + 41T^{2} \)
43 \( 1 - 2.80iT - 43T^{2} \)
47 \( 1 + 6.56iT - 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 - 5.84T + 59T^{2} \)
61 \( 1 + 9.59T + 61T^{2} \)
67 \( 1 - 9.59iT - 67T^{2} \)
71 \( 1 + 3.93T + 71T^{2} \)
73 \( 1 + 0.181iT - 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 - 3.70iT - 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 5.12iT - 97T^{2} \)
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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

−10.38442342750605554761583724998, −9.627633915481734854378373393137, −9.179910455339619722498137533278, −8.404181550933324197425675429334, −7.07653437764870589169498214084, −5.55422603075171724300782658399, −4.84487079998345566770788658942, −3.92293325722377959870442747333, −3.21086432491311283620442700998, −2.46742286386859498103090391580, 0.05544059176367377898148233595, 1.44127580800477130539781516623, 3.00728654470099911525739930709, 4.39158452325567070079208125054, 5.63773000498242270895299562042, 6.42284843942751487665839504352, 7.07494238518408462703238047304, 7.69927832410979883205834668884, 8.154610784982542605174052956309, 8.693538828818398888636607899153

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