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Properties

Label 2-1040-13.9-c1-0-9
Degree $2$
Conductor $1040$
Sign $0.522 - 0.852i$
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  + 5-s + (−1.5 − 2.59i)7-s + (1.5 + 2.59i)9-s + (−1.5 + 2.59i)11-s + (−3.5 + 0.866i)13-s + (2 + 3.46i)17-s + (3.5 + 6.06i)19-s + (−2 + 3.46i)23-s + 25-s + (4 − 6.92i)29-s + 10·31-s + (−1.5 − 2.59i)35-s + (−1.5 + 2.59i)37-s + (1 − 1.73i)41-s + (3 + 5.19i)43-s + ⋯
L(s)  = 1  + 0.447·5-s + (−0.566 − 0.981i)7-s + (0.5 + 0.866i)9-s + (−0.452 + 0.783i)11-s + (−0.970 + 0.240i)13-s + (0.485 + 0.840i)17-s + (0.802 + 1.39i)19-s + (−0.417 + 0.722i)23-s + 0.200·25-s + (0.742 − 1.28i)29-s + 1.79·31-s + (−0.253 − 0.439i)35-s + (−0.246 + 0.427i)37-s + (0.156 − 0.270i)41-s + (0.457 + 0.792i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.477618143\)
\(L(\frac12)\) \(\approx\) \(1.477618143\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (3.5 - 0.866i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8 + 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.900096912838660520424409182755, −9.828803287047650627160180626082, −8.067446308438749830899649893147, −7.67134612970128390849417748114, −6.77431777862100081282101146224, −5.79759526651364542510291869715, −4.76110135074437241919592529354, −3.96381096315618940633706225952, −2.62234119848420678874917353043, −1.43219173246558327857857503143, 0.70755527216395934544313643734, 2.61954838010685489440722603534, 3.11832468131058604268564139473, 4.74128519240617665761652416027, 5.47325668756707026351300984270, 6.43425235774601177400295620642, 7.10228348192114430904729985552, 8.264873616973091332588064471588, 9.174127321069994995867871803631, 9.633045943457412736749359149404

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