Properties

Label 2-1045-5.4-c1-0-15
Degree $2$
Conductor $1045$
Sign $-0.0288 - 0.999i$
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  + 1.19i·2-s − 2.62i·3-s + 0.580·4-s + (0.0644 + 2.23i)5-s + 3.12·6-s − 0.593i·7-s + 3.07i·8-s − 3.88·9-s + (−2.66 + 0.0767i)10-s − 11-s − 1.52i·12-s + 6.20i·13-s + 0.707·14-s + (5.86 − 0.169i)15-s − 2.50·16-s + 4.18i·17-s + ⋯
L(s)  = 1  + 0.842i·2-s − 1.51i·3-s + 0.290·4-s + (0.0288 + 0.999i)5-s + 1.27·6-s − 0.224i·7-s + 1.08i·8-s − 1.29·9-s + (−0.842 + 0.0242i)10-s − 0.301·11-s − 0.439i·12-s + 1.72i·13-s + 0.188·14-s + (1.51 − 0.0436i)15-s − 0.625·16-s + 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0288 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0288 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.0288 - 0.999i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.0288 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.572819633\)
\(L(\frac12)\) \(\approx\) \(1.572819633\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.0644 - 2.23i)T \)
11 \( 1 + T \)
19 \( 1 + T \)
good2 \( 1 - 1.19iT - 2T^{2} \)
3 \( 1 + 2.62iT - 3T^{2} \)
7 \( 1 + 0.593iT - 7T^{2} \)
13 \( 1 - 6.20iT - 13T^{2} \)
17 \( 1 - 4.18iT - 17T^{2} \)
23 \( 1 - 2.93iT - 23T^{2} \)
29 \( 1 - 3.45T + 29T^{2} \)
31 \( 1 + 6.32T + 31T^{2} \)
37 \( 1 - 1.18iT - 37T^{2} \)
41 \( 1 - 2.97T + 41T^{2} \)
43 \( 1 - 2.99iT - 43T^{2} \)
47 \( 1 + 7.94iT - 47T^{2} \)
53 \( 1 + 4.96iT - 53T^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 + 5.71T + 61T^{2} \)
67 \( 1 - 2.49iT - 67T^{2} \)
71 \( 1 - 6.87T + 71T^{2} \)
73 \( 1 + 6.57iT - 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 - 10.7iT - 83T^{2} \)
89 \( 1 + 6.38T + 89T^{2} \)
97 \( 1 - 7.26iT - 97T^{2} \)
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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

−10.23359350780052784962812480444, −8.939335188052240161829159984351, −8.029175015406453860359937270024, −7.40650367220298492048678036230, −6.70331922865792891912973503441, −6.43790540208841160615471229342, −5.45716213737517118225381230816, −3.85737884265958049786410053146, −2.39915871121573909540101295442, −1.75178711593464028137921263172, 0.67969249760368036503610383658, 2.47918037507947566295807095075, 3.36216070336536354018401584333, 4.31442452913386389676619475639, 5.17207350101692397880256264040, 5.85825743783092801340174717833, 7.37319655046602089194000311580, 8.395068584329088745335062321151, 9.229108905440709995855537706271, 9.839452418611490936488128618242

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