Properties

Label 2-1045-5.4-c1-0-24
Degree $2$
Conductor $1045$
Sign $0.934 + 0.356i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.71i·2-s + 2.46i·3-s − 5.38·4-s + (2.08 + 0.798i)5-s + 6.70·6-s − 0.267i·7-s + 9.18i·8-s − 3.08·9-s + (2.16 − 5.67i)10-s − 11-s − 13.2i·12-s − 4.26i·13-s − 0.726·14-s + (−1.96 + 5.15i)15-s + 14.1·16-s − 1.48i·17-s + ⋯
L(s)  = 1  − 1.92i·2-s + 1.42i·3-s − 2.69·4-s + (0.934 + 0.356i)5-s + 2.73·6-s − 0.101i·7-s + 3.24i·8-s − 1.02·9-s + (0.685 − 1.79i)10-s − 0.301·11-s − 3.83i·12-s − 1.18i·13-s − 0.194·14-s + (−0.508 + 1.33i)15-s + 3.54·16-s − 0.359i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.934 + 0.356i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.934 + 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.472566950\)
\(L(\frac12)\) \(\approx\) \(1.472566950\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.08 - 0.798i)T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 + 2.71iT - 2T^{2} \)
3 \( 1 - 2.46iT - 3T^{2} \)
7 \( 1 + 0.267iT - 7T^{2} \)
13 \( 1 + 4.26iT - 13T^{2} \)
17 \( 1 + 1.48iT - 17T^{2} \)
23 \( 1 - 8.83iT - 23T^{2} \)
29 \( 1 - 5.04T + 29T^{2} \)
31 \( 1 - 9.67T + 31T^{2} \)
37 \( 1 - 7.04iT - 37T^{2} \)
41 \( 1 + 2.49T + 41T^{2} \)
43 \( 1 - 5.23iT - 43T^{2} \)
47 \( 1 - 7.14iT - 47T^{2} \)
53 \( 1 - 8.90iT - 53T^{2} \)
59 \( 1 - 1.15T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 3.64iT - 67T^{2} \)
71 \( 1 - 8.54T + 71T^{2} \)
73 \( 1 + 6.43iT - 73T^{2} \)
79 \( 1 - 1.06T + 79T^{2} \)
83 \( 1 + 15.6iT - 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + 7.50iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.09093290513514705466778716882, −9.583466492223915874377971961079, −8.865764564617842333167410615834, −7.81633669076320514663375197336, −5.89631938505458729201420231320, −5.07721814684465535253403518366, −4.43793593817602044982921794808, −3.11989497835227190672921614843, −2.92698687227321744324407159833, −1.28137352644624084304599700006, 0.78749750211547144974719858188, 2.28194437690106112183338377696, 4.30659172632689315385842964790, 5.17852463542414272338765475921, 6.16498331850864744697745970666, 6.58997257808428960547268699803, 7.15894382777155399182892975537, 8.344178329724709949907628344123, 8.525139107043429617036986201401, 9.538695533248980377989574530181

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