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Properties

Label 2-1045-209.208-c1-0-70
Degree $2$
Conductor $1045$
Sign $0.986 - 0.166i$
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  + 2.69·2-s + 1.09i·3-s + 5.23·4-s + 5-s + 2.93i·6-s − 4.34i·7-s + 8.70·8-s + 1.80·9-s + 2.69·10-s + (−1.85 + 2.75i)11-s + 5.71i·12-s − 5.00·13-s − 11.6i·14-s + 1.09i·15-s + 12.9·16-s + 1.42i·17-s + ⋯
L(s)  = 1  + 1.90·2-s + 0.630i·3-s + 2.61·4-s + 0.447·5-s + 1.19i·6-s − 1.64i·7-s + 3.07·8-s + 0.602·9-s + 0.850·10-s + (−0.558 + 0.829i)11-s + 1.65i·12-s − 1.38·13-s − 3.12i·14-s + 0.281i·15-s + 3.23·16-s + 0.346i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.986 - 0.166i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.986 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(5.439518424\)
\(L(\frac12)\) \(\approx\) \(5.439518424\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.85 - 2.75i)T \)
19 \( 1 + (3.00 + 3.15i)T \)
good2 \( 1 - 2.69T + 2T^{2} \)
3 \( 1 - 1.09iT - 3T^{2} \)
7 \( 1 + 4.34iT - 7T^{2} \)
13 \( 1 + 5.00T + 13T^{2} \)
17 \( 1 - 1.42iT - 17T^{2} \)
23 \( 1 - 2.49T + 23T^{2} \)
29 \( 1 + 5.84T + 29T^{2} \)
31 \( 1 + 2.48iT - 31T^{2} \)
37 \( 1 - 6.42iT - 37T^{2} \)
41 \( 1 + 5.23T + 41T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 - 3.25T + 47T^{2} \)
53 \( 1 + 4.67iT - 53T^{2} \)
59 \( 1 + 1.28iT - 59T^{2} \)
61 \( 1 - 12.8iT - 61T^{2} \)
67 \( 1 + 5.61iT - 67T^{2} \)
71 \( 1 - 10.8iT - 71T^{2} \)
73 \( 1 + 8.64iT - 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 + 12.0iT - 83T^{2} \)
89 \( 1 + 5.26iT - 89T^{2} \)
97 \( 1 + 1.44iT - 97T^{2} \)
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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

−10.26319891815985672249292957600, −9.614044825995522003004761491130, −7.66573694014076488627999843918, −7.16440594621110626116825284752, −6.49249629542608448334528347538, −5.13930567329990687529757041773, −4.62783437919883191563313366298, −4.07998949344496599579211114874, −2.96429091074245723255002317523, −1.78059182069296360866083822562, 2.05053436633008025846283344156, 2.44730822817213694913802596969, 3.61914841313631698903478250132, 4.99539763715108236527062752666, 5.45129434626353688294129415425, 6.20064940155391828868294055814, 7.03709224522346202797309284104, 7.84693153749994399723758283552, 9.030862463492057916672911003629, 10.17636493353786849975344002296

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