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Properties

Label 2-1045-209.208-c1-0-22
Degree $2$
Conductor $1045$
Sign $0.741 + 0.671i$
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  − 1.23·2-s − 3.34i·3-s − 0.469·4-s + 5-s + 4.13i·6-s + 3.14i·7-s + 3.05·8-s − 8.18·9-s − 1.23·10-s + (3.08 + 1.21i)11-s + 1.57i·12-s − 4.68·13-s − 3.89i·14-s − 3.34i·15-s − 2.84·16-s + 0.133i·17-s + ⋯
L(s)  = 1  − 0.874·2-s − 1.93i·3-s − 0.234·4-s + 0.447·5-s + 1.68i·6-s + 1.19i·7-s + 1.08·8-s − 2.72·9-s − 0.391·10-s + (0.929 + 0.367i)11-s + 0.453i·12-s − 1.29·13-s − 1.04i·14-s − 0.863i·15-s − 0.710·16-s + 0.0324i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.741 + 0.671i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.741 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8824957531\)
\(L(\frac12)\) \(\approx\) \(0.8824957531\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
11 \( 1 + (-3.08 - 1.21i)T \)
19 \( 1 + (-4.08 - 1.53i)T \)
good2 \( 1 + 1.23T + 2T^{2} \)
3 \( 1 + 3.34iT - 3T^{2} \)
7 \( 1 - 3.14iT - 7T^{2} \)
13 \( 1 + 4.68T + 13T^{2} \)
17 \( 1 - 0.133iT - 17T^{2} \)
23 \( 1 - 2.70T + 23T^{2} \)
29 \( 1 - 9.38T + 29T^{2} \)
31 \( 1 - 9.11iT - 31T^{2} \)
37 \( 1 + 6.98iT - 37T^{2} \)
41 \( 1 + 0.185T + 41T^{2} \)
43 \( 1 - 1.06iT - 43T^{2} \)
47 \( 1 - 7.91T + 47T^{2} \)
53 \( 1 - 1.35iT - 53T^{2} \)
59 \( 1 - 3.19iT - 59T^{2} \)
61 \( 1 + 2.28iT - 61T^{2} \)
67 \( 1 + 0.902iT - 67T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 + 2.27T + 79T^{2} \)
83 \( 1 + 9.64iT - 83T^{2} \)
89 \( 1 - 9.59iT - 89T^{2} \)
97 \( 1 + 14.1iT - 97T^{2} \)
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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

−9.411638871192066470420881456839, −8.931046696296414602670497589588, −8.224414361595938896031908333523, −7.31196885910326091951024960821, −6.81066060781933934372661470033, −5.76117513548722638849009045872, −4.94393738000535526654754585214, −2.87572683392004827644607037428, −1.95550370721733244458805818393, −1.00688546755729064631987187168, 0.72274014871311676528330874421, 2.87192429237023354593643740809, 4.05498694876523951903611068892, 4.59279503053244287820611961440, 5.41072637247225581846091437511, 6.76182506167450006937542609114, 7.83000963886435675039199783499, 8.761062370904697289101663573699, 9.459854998676383724052433438896, 9.900459809526223011395394899210

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