Properties

Label 2-1045-5.4-c1-0-2
Degree $2$
Conductor $1045$
Sign $0.783 + 0.621i$
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  − 2.33i·2-s − 1.97i·3-s − 3.44·4-s + (−1.75 − 1.39i)5-s − 4.60·6-s + 4.79i·7-s + 3.37i·8-s − 0.889·9-s + (−3.24 + 4.08i)10-s + 11-s + 6.79i·12-s + 6.97i·13-s + 11.1·14-s + (−2.74 + 3.45i)15-s + 0.985·16-s − 1.90i·17-s + ⋯
L(s)  = 1  − 1.65i·2-s − 1.13i·3-s − 1.72·4-s + (−0.783 − 0.621i)5-s − 1.87·6-s + 1.81i·7-s + 1.19i·8-s − 0.296·9-s + (−1.02 + 1.29i)10-s + 0.301·11-s + 1.96i·12-s + 1.93i·13-s + 2.98·14-s + (−0.708 + 0.891i)15-s + 0.246·16-s − 0.461i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.783 + 0.621i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6771173295\)
\(L(\frac12)\) \(\approx\) \(0.6771173295\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.75 + 1.39i)T \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 + 2.33iT - 2T^{2} \)
3 \( 1 + 1.97iT - 3T^{2} \)
7 \( 1 - 4.79iT - 7T^{2} \)
13 \( 1 - 6.97iT - 13T^{2} \)
17 \( 1 + 1.90iT - 17T^{2} \)
23 \( 1 - 4.07iT - 23T^{2} \)
29 \( 1 + 2.15T + 29T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
37 \( 1 - 5.25iT - 37T^{2} \)
41 \( 1 + 4.11T + 41T^{2} \)
43 \( 1 - 2.83iT - 43T^{2} \)
47 \( 1 - 9.58iT - 47T^{2} \)
53 \( 1 + 4.41iT - 53T^{2} \)
59 \( 1 - 3.08T + 59T^{2} \)
61 \( 1 + 5.73T + 61T^{2} \)
67 \( 1 + 5.17iT - 67T^{2} \)
71 \( 1 + 8.60T + 71T^{2} \)
73 \( 1 - 2.32iT - 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 - 2.49iT - 83T^{2} \)
89 \( 1 + 8.26T + 89T^{2} \)
97 \( 1 + 0.339iT - 97T^{2} \)
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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

−9.526009767711792488729865336403, −9.228407756466503695506422148728, −8.533787542139354486127676354079, −7.46522176243418205866393470958, −6.48811730477252374785820741405, −5.26752841774101690697257273305, −4.31323254227256208878583317765, −3.21698082401179116172098218400, −2.01189964434016569447052999423, −1.50555830173936986421068117664, 0.31753370506964171560761569750, 3.49991200246673469304528492769, 3.93686050317386396778982415154, 4.84191721527761855220360573541, 5.74210120406686168494619016204, 6.91772457538705970328618183570, 7.40921859030110940284593777861, 8.031352345949509581918805470771, 8.965810813538767096120874703525, 10.12250547732720681516292103220

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