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Properties

Label 2-1045-5.4-c1-0-39
Degree $2$
Conductor $1045$
Sign $-0.0123 - 0.999i$
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  + 0.386i·2-s − 0.261i·3-s + 1.85·4-s + (0.0276 + 2.23i)5-s + 0.101·6-s + 4.13i·7-s + 1.48i·8-s + 2.93·9-s + (−0.864 + 0.0106i)10-s + 11-s − 0.484i·12-s − 0.244i·13-s − 1.59·14-s + (0.584 − 0.00721i)15-s + 3.12·16-s − 4.03i·17-s + ⋯
L(s)  = 1  + 0.273i·2-s − 0.150i·3-s + 0.925·4-s + (0.0123 + 0.999i)5-s + 0.0412·6-s + 1.56i·7-s + 0.526i·8-s + 0.977·9-s + (−0.273 + 0.00337i)10-s + 0.301·11-s − 0.139i·12-s − 0.0679i·13-s − 0.427·14-s + (0.150 − 0.00186i)15-s + 0.781·16-s − 0.979i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0123 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0123 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.0123 - 0.999i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.0123 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.167483431\)
\(L(\frac12)\) \(\approx\) \(2.167483431\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.0276 - 2.23i)T \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 - 0.386iT - 2T^{2} \)
3 \( 1 + 0.261iT - 3T^{2} \)
7 \( 1 - 4.13iT - 7T^{2} \)
13 \( 1 + 0.244iT - 13T^{2} \)
17 \( 1 + 4.03iT - 17T^{2} \)
23 \( 1 + 0.483iT - 23T^{2} \)
29 \( 1 + 5.24T + 29T^{2} \)
31 \( 1 + 9.76T + 31T^{2} \)
37 \( 1 + 1.52iT - 37T^{2} \)
41 \( 1 - 6.06T + 41T^{2} \)
43 \( 1 - 7.60iT - 43T^{2} \)
47 \( 1 + 9.01iT - 47T^{2} \)
53 \( 1 + 5.86iT - 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 - 7.58T + 61T^{2} \)
67 \( 1 - 2.72iT - 67T^{2} \)
71 \( 1 - 2.89T + 71T^{2} \)
73 \( 1 - 7.24iT - 73T^{2} \)
79 \( 1 + 5.91T + 79T^{2} \)
83 \( 1 - 4.94iT - 83T^{2} \)
89 \( 1 - 3.41T + 89T^{2} \)
97 \( 1 + 12.5iT - 97T^{2} \)
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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

−10.03833108860343331969871297410, −9.412591103458043219023274833878, −8.343538426523243489292475348622, −7.28086807396945428956018130339, −6.95924364324711648990412344335, −5.92985613238401843732738890570, −5.32223412763500918555444729690, −3.69200482978092491181627750056, −2.60767507685525407842597328554, −1.89967573597757831520699124628, 1.03174674809233467599607982822, 1.85394539043002327044788270834, 3.78596152869826261372192655560, 4.03556066316180141630528953237, 5.34285339433719826794488584729, 6.44148003224245190121830633789, 7.36463727029819467481791710967, 7.75146772120017889973316012315, 9.077905165716914300843170375948, 9.856461150785364015794671050110

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