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Properties

Label 2-1045-5.4-c1-0-38
Degree $2$
Conductor $1045$
Sign $-0.688 - 0.724i$
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  + 0.853i·2-s + 1.84i·3-s + 1.27·4-s + (1.54 + 1.62i)5-s − 1.57·6-s − 2.69i·7-s + 2.79i·8-s − 0.399·9-s + (−1.38 + 1.31i)10-s − 11-s + 2.34i·12-s − 3.98i·13-s + 2.30·14-s + (−2.98 + 2.84i)15-s + 0.159·16-s + 4.81i·17-s + ⋯
L(s)  = 1  + 0.603i·2-s + 1.06i·3-s + 0.635·4-s + (0.688 + 0.724i)5-s − 0.642·6-s − 1.01i·7-s + 0.987i·8-s − 0.133·9-s + (−0.437 + 0.415i)10-s − 0.301·11-s + 0.676i·12-s − 1.10i·13-s + 0.615·14-s + (−0.771 + 0.733i)15-s + 0.0398·16-s + 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.688 - 0.724i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.688 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.180747777\)
\(L(\frac12)\) \(\approx\) \(2.180747777\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.54 - 1.62i)T \)
11 \( 1 + T \)
19 \( 1 + T \)
good2 \( 1 - 0.853iT - 2T^{2} \)
3 \( 1 - 1.84iT - 3T^{2} \)
7 \( 1 + 2.69iT - 7T^{2} \)
13 \( 1 + 3.98iT - 13T^{2} \)
17 \( 1 - 4.81iT - 17T^{2} \)
23 \( 1 - 7.92iT - 23T^{2} \)
29 \( 1 + 0.916T + 29T^{2} \)
31 \( 1 + 5.21T + 31T^{2} \)
37 \( 1 + 1.51iT - 37T^{2} \)
41 \( 1 - 5.27T + 41T^{2} \)
43 \( 1 + 5.42iT - 43T^{2} \)
47 \( 1 - 2.11iT - 47T^{2} \)
53 \( 1 + 9.54iT - 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 - 5.11iT - 67T^{2} \)
71 \( 1 + 3.08T + 71T^{2} \)
73 \( 1 + 5.56iT - 73T^{2} \)
79 \( 1 + 0.412T + 79T^{2} \)
83 \( 1 + 6.93iT - 83T^{2} \)
89 \( 1 + 1.47T + 89T^{2} \)
97 \( 1 + 18.1iT - 97T^{2} \)
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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

−10.24331706817771814884247565267, −9.721133031211993875938254322112, −8.477775328332794968220202587557, −7.46986975867598774229600387735, −7.01517708088413258706537333080, −5.79857031350234465749069784938, −5.38333960710646880655888006879, −3.95486768992742668188343115410, −3.20368680498897338507279498740, −1.79983933300663721415201622571, 1.00362764946805033809207953155, 2.19347485082039170853312719423, 2.50759451432667006179070492897, 4.29572976718875385382429017186, 5.43773576869255632857609004151, 6.38744707764811574110749983922, 6.90906064661825153458370102215, 7.942510882954633761866115608636, 8.940802880774772666439923944778, 9.532526139258489152230728286969

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