Properties

Label 2-2240-16.13-c1-0-29
Degree $2$
Conductor $2240$
Sign $0.416 + 0.909i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
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  + (0.296 + 0.296i)3-s + (−0.707 + 0.707i)5-s + i·7-s − 2.82i·9-s + (−1.34 + 1.34i)11-s + (−0.680 − 0.680i)13-s − 0.419·15-s − 6.14·17-s + (3.27 + 3.27i)19-s + (−0.296 + 0.296i)21-s − 7.61i·23-s − 1.00i·25-s + (1.72 − 1.72i)27-s + (−2.41 − 2.41i)29-s + 9.71·31-s + ⋯
L(s)  = 1  + (0.171 + 0.171i)3-s + (−0.316 + 0.316i)5-s + 0.377i·7-s − 0.941i·9-s + (−0.405 + 0.405i)11-s + (−0.188 − 0.188i)13-s − 0.108·15-s − 1.49·17-s + (0.751 + 0.751i)19-s + (−0.0647 + 0.0647i)21-s − 1.58i·23-s − 0.200i·25-s + (0.332 − 0.332i)27-s + (−0.448 − 0.448i)29-s + 1.74·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.416 + 0.909i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.416 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.240737628\)
\(L(\frac12)\) \(\approx\) \(1.240737628\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-0.296 - 0.296i)T + 3iT^{2} \)
11 \( 1 + (1.34 - 1.34i)T - 11iT^{2} \)
13 \( 1 + (0.680 + 0.680i)T + 13iT^{2} \)
17 \( 1 + 6.14T + 17T^{2} \)
19 \( 1 + (-3.27 - 3.27i)T + 19iT^{2} \)
23 \( 1 + 7.61iT - 23T^{2} \)
29 \( 1 + (2.41 + 2.41i)T + 29iT^{2} \)
31 \( 1 - 9.71T + 31T^{2} \)
37 \( 1 + (-8.06 + 8.06i)T - 37iT^{2} \)
41 \( 1 + 2.60iT - 41T^{2} \)
43 \( 1 + (-1.00 + 1.00i)T - 43iT^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + (-6.93 + 6.93i)T - 53iT^{2} \)
59 \( 1 + (2.09 - 2.09i)T - 59iT^{2} \)
61 \( 1 + (2.63 + 2.63i)T + 61iT^{2} \)
67 \( 1 + (-4.31 - 4.31i)T + 67iT^{2} \)
71 \( 1 - 2.00iT - 71T^{2} \)
73 \( 1 + 0.827iT - 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + (-2.03 - 2.03i)T + 83iT^{2} \)
89 \( 1 + 4.46iT - 89T^{2} \)
97 \( 1 - 15.5T + 97T^{2} \)
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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

−8.808895593595896531792847799895, −8.256603341681896857128398725627, −7.32761144732623972675054213109, −6.53638779335167204077828354943, −5.90201697062114993929332294360, −4.70300067581669807533248845589, −4.07104003395995925403091873620, −2.98110402878730604550829821581, −2.20308742853234977595941726806, −0.45465377242019380839898458295, 1.16718778026461473334646542384, 2.41718533720466549879249223618, 3.31217880358116829658353397531, 4.58477399343516773002086839445, 4.93866506346079317383905773051, 6.09043668780844705060241998514, 6.99968386747285260076839764300, 7.72379044827333755913212270791, 8.267659104204352721184245877014, 9.132948628719353896387486608813

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