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Properties

Label 2-2240-16.13-c1-0-18
Degree $2$
Conductor $2240$
Sign $0.226 - 0.973i$
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  + (1.42 + 1.42i)3-s + (0.707 − 0.707i)5-s + i·7-s + 1.07i·9-s + (−1.52 + 1.52i)11-s + (3.04 + 3.04i)13-s + 2.01·15-s + 4.23·17-s + (4.95 + 4.95i)19-s + (−1.42 + 1.42i)21-s + 0.679i·23-s − 1.00i·25-s + (2.74 − 2.74i)27-s + (−6.70 − 6.70i)29-s − 3.91·31-s + ⋯
L(s)  = 1  + (0.824 + 0.824i)3-s + (0.316 − 0.316i)5-s + 0.377i·7-s + 0.359i·9-s + (−0.459 + 0.459i)11-s + (0.844 + 0.844i)13-s + 0.521·15-s + 1.02·17-s + (1.13 + 1.13i)19-s + (−0.311 + 0.311i)21-s + 0.141i·23-s − 0.200i·25-s + (0.527 − 0.527i)27-s + (−1.24 − 1.24i)29-s − 0.703·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.226 - 0.973i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.226 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.625878528\)
\(L(\frac12)\) \(\approx\) \(2.625878528\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 - 1.42i)T + 3iT^{2} \)
11 \( 1 + (1.52 - 1.52i)T - 11iT^{2} \)
13 \( 1 + (-3.04 - 3.04i)T + 13iT^{2} \)
17 \( 1 - 4.23T + 17T^{2} \)
19 \( 1 + (-4.95 - 4.95i)T + 19iT^{2} \)
23 \( 1 - 0.679iT - 23T^{2} \)
29 \( 1 + (6.70 + 6.70i)T + 29iT^{2} \)
31 \( 1 + 3.91T + 31T^{2} \)
37 \( 1 + (2.96 - 2.96i)T - 37iT^{2} \)
41 \( 1 + 7.19iT - 41T^{2} \)
43 \( 1 + (5.87 - 5.87i)T - 43iT^{2} \)
47 \( 1 - 9.08T + 47T^{2} \)
53 \( 1 + (-0.694 + 0.694i)T - 53iT^{2} \)
59 \( 1 + (-1.02 + 1.02i)T - 59iT^{2} \)
61 \( 1 + (-3.24 - 3.24i)T + 61iT^{2} \)
67 \( 1 + (-8.40 - 8.40i)T + 67iT^{2} \)
71 \( 1 - 9.32iT - 71T^{2} \)
73 \( 1 - 7.80iT - 73T^{2} \)
79 \( 1 + 7.64T + 79T^{2} \)
83 \( 1 + (-7.17 - 7.17i)T + 83iT^{2} \)
89 \( 1 - 12.8iT - 89T^{2} \)
97 \( 1 - 1.52T + 97T^{2} \)
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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

−9.354398989899721012913939038175, −8.511907152170438287816820895965, −7.916182100838345875896962063087, −6.97719328253282683226218816615, −5.77489742303247223963316525958, −5.33430173331217970519266192006, −4.06320386921420782354742259497, −3.63157242310413753942497458982, −2.51511592094007072567773827454, −1.42833845864083612372643570895, 0.882122194632564010771852306795, 1.94401931950711713038294715853, 3.15169266641847838208666629273, 3.42709399294718903890886689891, 5.11163692036187997337493424174, 5.67977183172039907570382227677, 6.79205520960021707602530229510, 7.46345319947582464516689156393, 7.930456947117590907126976597531, 8.806897895538774745715675641454

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