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Properties

Label 2-2240-5.4-c1-0-17
Degree $2$
Conductor $2240$
Sign $0.332 - 0.943i$
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  + 0.321i·3-s + (−2.10 − 0.742i)5-s + i·7-s + 2.89·9-s + 4.37·11-s + 5.86i·13-s + (0.238 − 0.678i)15-s − 4.85i·17-s − 7.75·19-s − 0.321·21-s + 1.35i·23-s + (3.89 + 3.13i)25-s + 1.89i·27-s − 0.539·29-s + 2.97·31-s + ⋯
L(s)  = 1  + 0.185i·3-s + (−0.943 − 0.332i)5-s + 0.377i·7-s + 0.965·9-s + 1.32·11-s + 1.62i·13-s + (0.0616 − 0.175i)15-s − 1.17i·17-s − 1.77·19-s − 0.0701·21-s + 0.282i·23-s + (0.779 + 0.626i)25-s + 0.364i·27-s − 0.100·29-s + 0.533·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.332 - 0.943i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.332 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.465114594\)
\(L(\frac12)\) \(\approx\) \(1.465114594\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 + 0.742i)T \)
7 \( 1 - iT \)
good3 \( 1 - 0.321iT - 3T^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
13 \( 1 - 5.86iT - 13T^{2} \)
17 \( 1 + 4.85iT - 17T^{2} \)
19 \( 1 + 7.75T + 19T^{2} \)
23 \( 1 - 1.35iT - 23T^{2} \)
29 \( 1 + 0.539T + 29T^{2} \)
31 \( 1 - 2.97T + 31T^{2} \)
37 \( 1 + 6.26iT - 37T^{2} \)
41 \( 1 - 2.64T + 41T^{2} \)
43 \( 1 - 4.64iT - 43T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 + 0.477iT - 53T^{2} \)
59 \( 1 - 7.75T + 59T^{2} \)
61 \( 1 + 7.57T + 61T^{2} \)
67 \( 1 - 3.79iT - 67T^{2} \)
71 \( 1 - 9.23T + 71T^{2} \)
73 \( 1 + 0.477iT - 73T^{2} \)
79 \( 1 - 1.88T + 79T^{2} \)
83 \( 1 - 15.2iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 13.6iT - 97T^{2} \)
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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

−9.260404950172923991703700960046, −8.568073278959866068350931822771, −7.57990284718845400759662936915, −6.82671387562040942010472874569, −6.32500345932592247120896873676, −4.88360662930132680278869394981, −4.25540321350106664884965938986, −3.80795192161127660813587342976, −2.30808724706677274581712361477, −1.17119617808064601953412422157, 0.59156304385804532084819310439, 1.83553166227328797718539638765, 3.25885984458710767186569355426, 4.02235572933946471316617325942, 4.55456795228821721050777603301, 5.96144058434331793148989215389, 6.68199428368724849918548145299, 7.25369623979183009850269469117, 8.314724761261167874406036377751, 8.451498976419150457144127108582

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