Properties

Label 2-2240-5.4-c1-0-1
Degree $2$
Conductor $2240$
Sign $-0.936 + 0.350i$
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  + 0.746i·3-s + (0.782 + 2.09i)5-s i·7-s + 2.44·9-s − 5.90·11-s − 3.20i·13-s + (−1.56 + 0.584i)15-s + 2.14i·17-s − 3.56·19-s + 0.746·21-s − 3.75i·23-s + (−3.77 + 3.27i)25-s + 4.06i·27-s − 6.61·29-s − 5.79·31-s + ⋯
L(s)  = 1  + 0.431i·3-s + (0.350 + 0.936i)5-s − 0.377i·7-s + 0.814·9-s − 1.78·11-s − 0.890i·13-s + (−0.403 + 0.151i)15-s + 0.521i·17-s − 0.818·19-s + 0.163·21-s − 0.782i·23-s + (−0.754 + 0.655i)25-s + 0.782i·27-s − 1.22·29-s − 1.04·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.936 + 0.350i$
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: \((2,\ 2240,\ (\ :1/2),\ -0.936 + 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2823503060\)
\(L(\frac12)\) \(\approx\) \(0.2823503060\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.782 - 2.09i)T \)
7 \( 1 + iT \)
good3 \( 1 - 0.746iT - 3T^{2} \)
11 \( 1 + 5.90T + 11T^{2} \)
13 \( 1 + 3.20iT - 13T^{2} \)
17 \( 1 - 2.14iT - 17T^{2} \)
19 \( 1 + 3.56T + 19T^{2} \)
23 \( 1 + 3.75iT - 23T^{2} \)
29 \( 1 + 6.61T + 29T^{2} \)
31 \( 1 + 5.79T + 31T^{2} \)
37 \( 1 - 0.623iT - 37T^{2} \)
41 \( 1 + 5.43T + 41T^{2} \)
43 \( 1 - 12.6iT - 43T^{2} \)
47 \( 1 + 4.31iT - 47T^{2} \)
53 \( 1 + 2.11iT - 53T^{2} \)
59 \( 1 + 7.01T + 59T^{2} \)
61 \( 1 + 1.86T + 61T^{2} \)
67 \( 1 - 6.88iT - 67T^{2} \)
71 \( 1 + 1.81T + 71T^{2} \)
73 \( 1 - 6.11iT - 73T^{2} \)
79 \( 1 - 4.35T + 79T^{2} \)
83 \( 1 + 13.1iT - 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 - 9.65iT - 97T^{2} \)
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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

−9.845162315825135807162335351961, −8.642327713093344748941721403865, −7.75370114138566519043443208847, −7.29842736880156237154398455962, −6.31948926001850256246640147268, −5.51629900099346898789004176550, −4.69159090752835442124158267744, −3.68014250802647555516235901341, −2.84912586012503224045703010444, −1.84636606705999207980311595319, 0.087493497970473807217410566483, 1.69769941474920271424875823702, 2.29301202383612851208860604719, 3.74066288718324149714555183431, 4.77701414539935015817975606291, 5.34728243601630313347802963024, 6.17122011476938291353381197077, 7.27202555634339710497153289233, 7.71649559561845180802532918361, 8.674399268955481567833069234533

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