Properties

Label 2-2240-16.5-c1-0-31
Degree $2$
Conductor $2240$
Sign $-0.0169 + 0.999i$
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  + (−1.63 + 1.63i)3-s + (−0.707 − 0.707i)5-s i·7-s − 2.32i·9-s + (0.230 + 0.230i)11-s + (−1.57 + 1.57i)13-s + 2.30·15-s + 4.13·17-s + (−0.553 + 0.553i)19-s + (1.63 + 1.63i)21-s + 0.651i·23-s + 1.00i·25-s + (−1.09 − 1.09i)27-s + (−5.41 + 5.41i)29-s − 4.74·31-s + ⋯
L(s)  = 1  + (−0.942 + 0.942i)3-s + (−0.316 − 0.316i)5-s − 0.377i·7-s − 0.776i·9-s + (0.0694 + 0.0694i)11-s + (−0.437 + 0.437i)13-s + 0.595·15-s + 1.00·17-s + (−0.127 + 0.127i)19-s + (0.356 + 0.356i)21-s + 0.135i·23-s + 0.200i·25-s + (−0.211 − 0.211i)27-s + (−1.00 + 1.00i)29-s − 0.851·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0169 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0169 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.0169 + 0.999i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.0169 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3674548778\)
\(L(\frac12)\) \(\approx\) \(0.3674548778\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 - 1.63i)T - 3iT^{2} \)
11 \( 1 + (-0.230 - 0.230i)T + 11iT^{2} \)
13 \( 1 + (1.57 - 1.57i)T - 13iT^{2} \)
17 \( 1 - 4.13T + 17T^{2} \)
19 \( 1 + (0.553 - 0.553i)T - 19iT^{2} \)
23 \( 1 - 0.651iT - 23T^{2} \)
29 \( 1 + (5.41 - 5.41i)T - 29iT^{2} \)
31 \( 1 + 4.74T + 31T^{2} \)
37 \( 1 + (-4.34 - 4.34i)T + 37iT^{2} \)
41 \( 1 + 6.29iT - 41T^{2} \)
43 \( 1 + (1.77 + 1.77i)T + 43iT^{2} \)
47 \( 1 - 2.06T + 47T^{2} \)
53 \( 1 + (3.94 + 3.94i)T + 53iT^{2} \)
59 \( 1 + (10.2 + 10.2i)T + 59iT^{2} \)
61 \( 1 + (3.32 - 3.32i)T - 61iT^{2} \)
67 \( 1 + (-10.7 + 10.7i)T - 67iT^{2} \)
71 \( 1 - 2.53iT - 71T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 + 6.25T + 79T^{2} \)
83 \( 1 + (4.72 - 4.72i)T - 83iT^{2} \)
89 \( 1 + 9.25iT - 89T^{2} \)
97 \( 1 + 2.47T + 97T^{2} \)
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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

−9.117088750092971246287437472320, −7.985368164299426544220889782339, −7.32287021330684338910505656124, −6.36075926746489942612638544172, −5.43247165964044757379381124233, −4.92923685294439739998256167619, −4.06743428987168816345261991540, −3.34639171506722317193738473638, −1.69385274718689124714443971387, −0.16565133958671237831873712479, 1.09016063082199111789902526058, 2.31804121821414194552385862978, 3.39188991478519306724337305024, 4.53478556397849056273287121081, 5.70743495029049353154736887691, 5.90249092855766320554741175204, 6.98211962687963568519472245346, 7.53631240434769475584214329224, 8.173627775600571785087654140330, 9.300622499946275765776227046916

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