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Properties

Label 2-2240-28.27-c1-0-58
Degree $2$
Conductor $2240$
Sign $0.120 + 0.992i$
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  + 2.58·3-s − i·5-s + (−0.319 − 2.62i)7-s + 3.66·9-s − 0.0961i·11-s − 2.44i·13-s − 2.58i·15-s − 2.17i·17-s + 2.11·19-s + (−0.823 − 6.77i)21-s − 2.75i·23-s − 25-s + 1.70·27-s − 6.89·29-s + 2.00·31-s + ⋯
L(s)  = 1  + 1.49·3-s − 0.447i·5-s + (−0.120 − 0.992i)7-s + 1.22·9-s − 0.0290i·11-s − 0.676i·13-s − 0.666i·15-s − 0.527i·17-s + 0.484·19-s + (−0.179 − 1.47i)21-s − 0.573i·23-s − 0.200·25-s + 0.328·27-s − 1.28·29-s + 0.360·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.120 + 0.992i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.120 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.801213536\)
\(L(\frac12)\) \(\approx\) \(2.801213536\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (0.319 + 2.62i)T \)
good3 \( 1 - 2.58T + 3T^{2} \)
11 \( 1 + 0.0961iT - 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + 2.17iT - 17T^{2} \)
19 \( 1 - 2.11T + 19T^{2} \)
23 \( 1 + 2.75iT - 23T^{2} \)
29 \( 1 + 6.89T + 29T^{2} \)
31 \( 1 - 2.00T + 31T^{2} \)
37 \( 1 - 3.95T + 37T^{2} \)
41 \( 1 - 6.44iT - 41T^{2} \)
43 \( 1 + 6.88iT - 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 - 13.6T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 8.32iT - 61T^{2} \)
67 \( 1 + 0.286iT - 67T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + 4.38iT - 73T^{2} \)
79 \( 1 - 9.94iT - 79T^{2} \)
83 \( 1 - 9.06T + 83T^{2} \)
89 \( 1 + 4.14iT - 89T^{2} \)
97 \( 1 + 16.3iT - 97T^{2} \)
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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

−8.696842698062699383986450266715, −8.180911655975605669069502441211, −7.48790786949011144474278527838, −6.85956477941589409448998785005, −5.61572567597757262437468230617, −4.62355814623978306160051625651, −3.76293328808167954518523067849, −3.10021593902598743073015884419, −2.06111020902435674154206866015, −0.77946365596123461164777714780, 1.75749402383719701026292961945, 2.47053622760055939802429799488, 3.33653401470938881571964516187, 4.02948072312863581723291397339, 5.26871658423634893300958835985, 6.16331548328165270600543653998, 7.07767325196824876079867932941, 7.83896539520027403893392778804, 8.469453682961162372483995252820, 9.267645059424417718589437878659

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