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Properties

Label 2-1045-5.4-c1-0-34
Degree $2$
Conductor $1045$
Sign $0.994 - 0.103i$
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  + 0.131i·2-s − 1.44i·3-s + 1.98·4-s + (−2.22 + 0.230i)5-s + 0.189·6-s + 0.456i·7-s + 0.524i·8-s + 0.925·9-s + (−0.0303 − 0.292i)10-s + 11-s − 2.85i·12-s + 5.24i·13-s − 0.0601·14-s + (0.331 + 3.20i)15-s + 3.89·16-s + 6.95i·17-s + ⋯
L(s)  = 1  + 0.0930i·2-s − 0.831i·3-s + 0.991·4-s + (−0.994 + 0.103i)5-s + 0.0773·6-s + 0.172i·7-s + 0.185i·8-s + 0.308·9-s + (−0.00958 − 0.0925i)10-s + 0.301·11-s − 0.824i·12-s + 1.45i·13-s − 0.0160·14-s + (0.0857 + 0.827i)15-s + 0.974·16-s + 1.68i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.994 - 0.103i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.994 - 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.847193889\)
\(L(\frac12)\) \(\approx\) \(1.847193889\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.22 - 0.230i)T \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 - 0.131iT - 2T^{2} \)
3 \( 1 + 1.44iT - 3T^{2} \)
7 \( 1 - 0.456iT - 7T^{2} \)
13 \( 1 - 5.24iT - 13T^{2} \)
17 \( 1 - 6.95iT - 17T^{2} \)
23 \( 1 + 3.13iT - 23T^{2} \)
29 \( 1 - 4.35T + 29T^{2} \)
31 \( 1 - 2.59T + 31T^{2} \)
37 \( 1 - 0.922iT - 37T^{2} \)
41 \( 1 + 3.30T + 41T^{2} \)
43 \( 1 + 0.870iT - 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 - 8.75iT - 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 3.21T + 61T^{2} \)
67 \( 1 + 1.23iT - 67T^{2} \)
71 \( 1 - 8.08T + 71T^{2} \)
73 \( 1 - 2.75iT - 73T^{2} \)
79 \( 1 + 3.24T + 79T^{2} \)
83 \( 1 + 4.84iT - 83T^{2} \)
89 \( 1 - 5.97T + 89T^{2} \)
97 \( 1 + 9.50iT - 97T^{2} \)
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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

−10.16075972293688939463807561789, −8.785420806725388005079129999571, −8.120839844334275820871485995735, −7.28714360025517018988760111738, −6.67071590982194149182856699473, −6.13570067311835092299681748512, −4.54080846976218467180262740559, −3.65512019260960383040604090726, −2.30852590378305600227902621241, −1.33733920486374301276467875477, 0.964290341612790503501445626874, 2.86593118262433756615389166489, 3.50155229196107191730881426161, 4.61729359730364599595210369858, 5.42551899459171290632562298265, 6.71459933760684991233041353153, 7.46405247996009707593226359259, 8.054521749977251308584327772892, 9.268515099740047496327974899202, 10.04018833194300805622735733425

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