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Properties

Label 2-1045-209.208-c1-0-74
Degree $2$
Conductor $1045$
Sign $-0.286 + 0.958i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.286 + 0.958i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.286 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.131615375\)
\(L(\frac12)\) \(\approx\) \(4.131615375\)
\(L(\frac{3}{2})\) 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

−9.794073429173681177147554290947, −8.543024462159007418179227674542, −7.47165439483657688106325194176, −6.92620591690648773639385294954, −6.34400244765960491599128792137, −5.21936113325379204151119703350, −4.78045836544641145377355290599, −2.88743938271776834320906264331, −2.66155136936236307265591974153, −1.14039744606239358203360440541, 2.42678991666562031190056236744, 3.24949166880804371499809954444, 4.12221137548614280606061343167, 4.89950069033853568860117772975, 5.59822721105682421635571126200, 6.09588086208846772552333714994, 7.50346992140843877703061179604, 8.703906518368934327030639045212, 9.525081202236842309234063325062, 10.51227021375166871148596515622

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