Properties

Label 2-1045-209.208-c1-0-51
Degree $2$
Conductor $1045$
Sign $0.685 + 0.727i$
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  − 2.36·2-s + 2.84i·3-s + 3.57·4-s + 5-s − 6.72i·6-s − 0.816i·7-s − 3.71·8-s − 5.11·9-s − 2.36·10-s + (−3.22 + 0.764i)11-s + 10.1i·12-s + 0.847·13-s + 1.92i·14-s + 2.84i·15-s + 1.62·16-s − 4.07i·17-s + ⋯
L(s)  = 1  − 1.66·2-s + 1.64i·3-s + 1.78·4-s + 0.447·5-s − 2.74i·6-s − 0.308i·7-s − 1.31·8-s − 1.70·9-s − 0.746·10-s + (−0.973 + 0.230i)11-s + 2.93i·12-s + 0.234·13-s + 0.515i·14-s + 0.735i·15-s + 0.406·16-s − 0.987i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.685 + 0.727i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.685 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3308279151\)
\(L(\frac12)\) \(\approx\) \(0.3308279151\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.22 - 0.764i)T \)
19 \( 1 + (2.17 + 3.77i)T \)
good2 \( 1 + 2.36T + 2T^{2} \)
3 \( 1 - 2.84iT - 3T^{2} \)
7 \( 1 + 0.816iT - 7T^{2} \)
13 \( 1 - 0.847T + 13T^{2} \)
17 \( 1 + 4.07iT - 17T^{2} \)
23 \( 1 - 6.65T + 23T^{2} \)
29 \( 1 + 7.23T + 29T^{2} \)
31 \( 1 + 5.91iT - 31T^{2} \)
37 \( 1 + 9.96iT - 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 - 5.14iT - 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + 1.18iT - 53T^{2} \)
59 \( 1 - 1.07iT - 59T^{2} \)
61 \( 1 + 13.4iT - 61T^{2} \)
67 \( 1 + 9.83iT - 67T^{2} \)
71 \( 1 + 4.42iT - 71T^{2} \)
73 \( 1 - 4.87iT - 73T^{2} \)
79 \( 1 - 5.76T + 79T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 + 5.52iT - 89T^{2} \)
97 \( 1 - 13.9iT - 97T^{2} \)
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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

−9.490343470330806101433095722754, −9.437609073527839574081806693923, −8.520360248257569357871490046251, −7.58856792013841060294118861789, −6.71194198461996352149205994731, −5.38759174968303517760462080515, −4.67886278444012460391732803836, −3.29683539161338379428162062581, −2.21594827426533810333061694631, −0.26310207880645412734015207024, 1.30433353228533355930398307613, 1.96560823561339406180265264763, 3.05682052469704605457588060665, 5.34299444584273432548571753636, 6.27061798899249473020801919595, 6.93391822676179266514973287235, 7.68355983066743793317003142419, 8.513526603154169276224679320106, 8.717874286373257550106049920252, 10.05509105339275964244567984881

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