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Properties

Label 2-1040-13.10-c1-0-27
Degree $2$
Conductor $1040$
Sign $-0.916 - 0.400i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + (−0.913 − 1.58i)3-s − i·5-s + (−3.45 − 1.99i)7-s + (−0.168 + 0.292i)9-s + (4.24 − 2.44i)11-s + (−2.87 + 2.17i)13-s + (−1.58 + 0.913i)15-s + (3.31 − 5.74i)17-s + (−1.81 − 1.04i)19-s + 7.29i·21-s + (0.495 + 0.858i)23-s − 25-s − 4.86·27-s + (−2.92 − 5.06i)29-s + 10.8i·31-s + ⋯
L(s)  = 1  + (−0.527 − 0.913i)3-s − 0.447i·5-s + (−1.30 − 0.754i)7-s + (−0.0562 + 0.0973i)9-s + (1.27 − 0.738i)11-s + (−0.797 + 0.603i)13-s + (−0.408 + 0.235i)15-s + (0.803 − 1.39i)17-s + (−0.416 − 0.240i)19-s + 1.59i·21-s + (0.103 + 0.178i)23-s − 0.200·25-s − 0.936·27-s + (−0.542 − 0.939i)29-s + 1.94i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.916 - 0.400i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.916 - 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6634151675\)
\(L(\frac12)\) \(\approx\) \(0.6634151675\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + iT \)
13 \( 1 + (2.87 - 2.17i)T \)
good3 \( 1 + (0.913 + 1.58i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.45 + 1.99i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.24 + 2.44i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.31 + 5.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.81 + 1.04i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.495 - 0.858i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.92 + 5.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.8iT - 31T^{2} \)
37 \( 1 + (2.85 - 1.64i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.48 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.567 - 0.983i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.61iT - 47T^{2} \)
53 \( 1 + 0.549T + 53T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.685 + 1.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.13 - 1.81i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.94iT - 73T^{2} \)
79 \( 1 + 2.55T + 79T^{2} \)
83 \( 1 + 16.4iT - 83T^{2} \)
89 \( 1 + (-2.98 + 1.72i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.0 + 5.82i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.514249013129501463969168288277, −8.751636526282466768894741563384, −7.42228287245363675183798550896, −6.84858920458683650904305128105, −6.35450460182480020008442710771, −5.26275014652475945096688490932, −4.04105160776796288436534292859, −3.09483790740876179422305178871, −1.38148212773777885834887599785, −0.33341306123771513958370125614, 2.08288932461747833220507719732, 3.44880481134969099183757233753, 4.10555512095758569129283961255, 5.35249542051222804583813166062, 6.05914491647826902255692741782, 6.83266759830088009384110663874, 7.87985563023681006515002414007, 9.115784637129497639827388176203, 9.722035438582974237178481209461, 10.21968589887093501428809098629

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