Properties

Label 2-2240-5.4-c1-0-51
Degree $2$
Conductor $2240$
Sign $0.0685 + 0.997i$
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  − 1.63i·3-s + (2.23 − 0.153i)5-s i·7-s + 0.328·9-s + 1.24·11-s + 4.20i·13-s + (−0.250 − 3.64i)15-s − 3.39i·17-s − 6.46·19-s − 1.63·21-s − 2.15i·23-s + (4.95 − 0.683i)25-s − 5.44i·27-s + 3.96·29-s + 10.0·31-s + ⋯
L(s)  = 1  − 0.943i·3-s + (0.997 − 0.0685i)5-s − 0.377i·7-s + 0.109·9-s + 0.374·11-s + 1.16i·13-s + (−0.0646 − 0.941i)15-s − 0.823i·17-s − 1.48·19-s − 0.356·21-s − 0.449i·23-s + (0.990 − 0.136i)25-s − 1.04i·27-s + 0.736·29-s + 1.80·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0685 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0685 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.0685 + 0.997i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.0685 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.256394352\)
\(L(\frac12)\) \(\approx\) \(2.256394352\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.153i)T \)
7 \( 1 + iT \)
good3 \( 1 + 1.63iT - 3T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 - 4.20iT - 13T^{2} \)
17 \( 1 + 3.39iT - 17T^{2} \)
19 \( 1 + 6.46T + 19T^{2} \)
23 \( 1 + 2.15iT - 23T^{2} \)
29 \( 1 - 3.96T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 + 6.76iT - 37T^{2} \)
41 \( 1 + 0.131T + 41T^{2} \)
43 \( 1 + 7.40iT - 43T^{2} \)
47 \( 1 + 4.82iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 6.33T + 61T^{2} \)
67 \( 1 - 2.65iT - 67T^{2} \)
71 \( 1 + 0.754T + 71T^{2} \)
73 \( 1 + 6.03iT - 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 - 0.914iT - 97T^{2} \)
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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

−8.862986727793099764916592289153, −8.076442541891759818842071264129, −6.98304587080044012153898652919, −6.66548242063586794713623410355, −6.02175298698193310131499981629, −4.77719751250022816186757002395, −4.12903927759461349683450556110, −2.56901023441228466631142828595, −1.90505449819297633654915153794, −0.845740207588039770570828465038, 1.34175663162310539641584941773, 2.54925545753833756854625451552, 3.46177990392181426763802590546, 4.54530649729413500198082078154, 5.10698126975598583592349841184, 6.22289669946675899285091200314, 6.46774105555419535584799179843, 7.930015064653298795139997901927, 8.560004779355853337334432985880, 9.355116336164562434436725369597

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