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Properties

Label 2-2240-16.13-c1-0-33
Degree $2$
Conductor $2240$
Sign $0.444 + 0.895i$
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  + (0.576 + 0.576i)3-s + (0.707 − 0.707i)5-s + i·7-s − 2.33i·9-s + (−2.93 + 2.93i)11-s + (−1.89 − 1.89i)13-s + 0.815·15-s + 1.43·17-s + (−1.45 − 1.45i)19-s + (−0.576 + 0.576i)21-s − 8.06i·23-s − 1.00i·25-s + (3.07 − 3.07i)27-s + (2.86 + 2.86i)29-s + 1.31·31-s + ⋯
L(s)  = 1  + (0.332 + 0.332i)3-s + (0.316 − 0.316i)5-s + 0.377i·7-s − 0.778i·9-s + (−0.883 + 0.883i)11-s + (−0.524 − 0.524i)13-s + 0.210·15-s + 0.348·17-s + (−0.333 − 0.333i)19-s + (−0.125 + 0.125i)21-s − 1.68i·23-s − 0.200i·25-s + (0.591 − 0.591i)27-s + (0.531 + 0.531i)29-s + 0.235·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.444 + 0.895i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.661086695\)
\(L(\frac12)\) \(\approx\) \(1.661086695\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.576 - 0.576i)T + 3iT^{2} \)
11 \( 1 + (2.93 - 2.93i)T - 11iT^{2} \)
13 \( 1 + (1.89 + 1.89i)T + 13iT^{2} \)
17 \( 1 - 1.43T + 17T^{2} \)
19 \( 1 + (1.45 + 1.45i)T + 19iT^{2} \)
23 \( 1 + 8.06iT - 23T^{2} \)
29 \( 1 + (-2.86 - 2.86i)T + 29iT^{2} \)
31 \( 1 - 1.31T + 31T^{2} \)
37 \( 1 + (-6.64 + 6.64i)T - 37iT^{2} \)
41 \( 1 + 8.42iT - 41T^{2} \)
43 \( 1 + (-1.52 + 1.52i)T - 43iT^{2} \)
47 \( 1 - 7.14T + 47T^{2} \)
53 \( 1 + (4.13 - 4.13i)T - 53iT^{2} \)
59 \( 1 + (-5.34 + 5.34i)T - 59iT^{2} \)
61 \( 1 + (3.30 + 3.30i)T + 61iT^{2} \)
67 \( 1 + (5.94 + 5.94i)T + 67iT^{2} \)
71 \( 1 + 4.72iT - 71T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + (-0.898 - 0.898i)T + 83iT^{2} \)
89 \( 1 + 11.5iT - 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−8.965617714919335389825964196873, −8.261908440508251675528660880658, −7.41252326994433911450057238667, −6.55253390353505934454002019674, −5.66707353826914517673339044593, −4.85080672416075925902496520086, −4.12736463660877389622944220132, −2.86702744483935290145437081595, −2.24792769586678122501793399474, −0.56076928006889474670203952983, 1.30398512428032760834863978590, 2.45975004711287036211004009174, 3.15816752315555922309041847839, 4.35953053998641823378381982278, 5.27644532127927058785862313751, 6.03619141003594055328368809193, 6.94766526978050931186556782270, 7.892000264744571008566926081133, 8.022188821330279597253386608063, 9.207556841036876678794761464831

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