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Properties

Label 2-2240-16.5-c1-0-26
Degree $2$
Conductor $2240$
Sign $0.911 + 0.410i$
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  + (−0.693 + 0.693i)3-s + (−0.707 − 0.707i)5-s − i·7-s + 2.03i·9-s + (0.309 + 0.309i)11-s + (3.90 − 3.90i)13-s + 0.980·15-s + 0.135·17-s + (−2.90 + 2.90i)19-s + (0.693 + 0.693i)21-s + 0.323i·23-s + 1.00i·25-s + (−3.49 − 3.49i)27-s + (5.89 − 5.89i)29-s − 6.72·31-s + ⋯
L(s)  = 1  + (−0.400 + 0.400i)3-s + (−0.316 − 0.316i)5-s − 0.377i·7-s + 0.679i·9-s + (0.0933 + 0.0933i)11-s + (1.08 − 1.08i)13-s + 0.253·15-s + 0.0329·17-s + (−0.667 + 0.667i)19-s + (0.151 + 0.151i)21-s + 0.0674i·23-s + 0.200i·25-s + (−0.672 − 0.672i)27-s + (1.09 − 1.09i)29-s − 1.20·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.911 + 0.410i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.911 + 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.354230486\)
\(L(\frac12)\) \(\approx\) \(1.354230486\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.693 - 0.693i)T - 3iT^{2} \)
11 \( 1 + (-0.309 - 0.309i)T + 11iT^{2} \)
13 \( 1 + (-3.90 + 3.90i)T - 13iT^{2} \)
17 \( 1 - 0.135T + 17T^{2} \)
19 \( 1 + (2.90 - 2.90i)T - 19iT^{2} \)
23 \( 1 - 0.323iT - 23T^{2} \)
29 \( 1 + (-5.89 + 5.89i)T - 29iT^{2} \)
31 \( 1 + 6.72T + 31T^{2} \)
37 \( 1 + (-3.99 - 3.99i)T + 37iT^{2} \)
41 \( 1 + 3.21iT - 41T^{2} \)
43 \( 1 + (1.58 + 1.58i)T + 43iT^{2} \)
47 \( 1 - 0.680T + 47T^{2} \)
53 \( 1 + (-1.42 - 1.42i)T + 53iT^{2} \)
59 \( 1 + (-3.10 - 3.10i)T + 59iT^{2} \)
61 \( 1 + (-3.16 + 3.16i)T - 61iT^{2} \)
67 \( 1 + (-9.61 + 9.61i)T - 67iT^{2} \)
71 \( 1 + 0.0650iT - 71T^{2} \)
73 \( 1 + 3.41iT - 73T^{2} \)
79 \( 1 - 9.73T + 79T^{2} \)
83 \( 1 + (-11.8 + 11.8i)T - 83iT^{2} \)
89 \( 1 - 1.93iT - 89T^{2} \)
97 \( 1 - 8.46T + 97T^{2} \)
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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

−8.900001411396711813837008164381, −8.085547165007464175515576226205, −7.71805275389104558210698690594, −6.49403125741315449449842431341, −5.77742603378401882149511639683, −4.97986087238491824897556086166, −4.14879028319611776101100555640, −3.39704121418053145915898600973, −2.02653289832579398982713852868, −0.64781378419305781598452395577, 0.950306506777499214248204095580, 2.16291944891748210140862799904, 3.39564277490620819837261398447, 4.12466123006335629040211662611, 5.20514091596028561878803473575, 6.23062963613030120940853480077, 6.62330112280584893654999821400, 7.34000577264581136260696540648, 8.506764446662626138316470841200, 8.927444644641278068397600864521

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