Properties

Label 2-1040-65.29-c1-0-19
Degree $2$
Conductor $1040$
Sign $0.921 - 0.387i$
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  + (1.86 − 1.07i)3-s + (−0.817 + 2.08i)5-s + (2.54 + 1.46i)7-s + (0.817 − 1.41i)9-s + (−0.317 − 0.550i)11-s + (3.60 + 0.0716i)13-s + (0.716 + 4.76i)15-s + (−1.05 − 0.611i)17-s + (−0.682 + 1.18i)19-s + 6.32·21-s + (−1.86 + 1.07i)23-s + (−3.66 − 3.40i)25-s + 2.93i·27-s + (1.5 + 2.59i)29-s + 8.96·31-s + ⋯
L(s)  = 1  + (1.07 − 0.621i)3-s + (−0.365 + 0.930i)5-s + (0.961 + 0.555i)7-s + (0.272 − 0.472i)9-s + (−0.0957 − 0.165i)11-s + (0.999 + 0.0198i)13-s + (0.184 + 1.22i)15-s + (−0.257 − 0.148i)17-s + (−0.156 + 0.271i)19-s + 1.38·21-s + (−0.388 + 0.224i)23-s + (−0.732 − 0.680i)25-s + 0.565i·27-s + (0.278 + 0.482i)29-s + 1.60·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.921 - 0.387i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.425900546\)
\(L(\frac12)\) \(\approx\) \(2.425900546\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.817 - 2.08i)T \)
13 \( 1 + (-3.60 - 0.0716i)T \)
good3 \( 1 + (-1.86 + 1.07i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.54 - 1.46i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.317 + 0.550i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.05 + 0.611i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.682 - 1.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.86 - 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.96T + 31T^{2} \)
37 \( 1 + (1.05 - 0.611i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.98 - 8.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.18 + 0.683i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.16iT - 47T^{2} \)
53 \( 1 - 0.642iT - 53T^{2} \)
59 \( 1 + (3.79 - 6.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.95 + 4.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.31 + 2.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 - 1.03T + 79T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (6.27 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.8 + 7.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.953647688347304413841770635319, −8.778691458454208604494096654353, −8.263303299093143636986917424214, −7.72461681843562940934518897629, −6.75685174091565718361770040412, −5.90322614476755604027719743935, −4.59227076241103073550690799376, −3.41341640649135322335512125935, −2.60095601349640785186624101007, −1.59190360034883034003677036905, 1.11414172672724402846744736124, 2.50926166875816843744106645503, 3.93542889419720329488326880805, 4.26583182780408302095595982062, 5.28445095865193036377979268123, 6.55155093182416885937986654989, 7.940434099913182340201915056906, 8.178228696679199024133733716153, 8.934151394713604442889973708854, 9.682327226104143294031229525016

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