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Properties

Label 2-2240-16.5-c1-0-23
Degree $2$
Conductor $2240$
Sign $0.988 + 0.152i$
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  + (0.605 − 0.605i)3-s + (−0.707 − 0.707i)5-s − i·7-s + 2.26i·9-s + (0.812 + 0.812i)11-s + (−2.79 + 2.79i)13-s − 0.856·15-s + 4.69·17-s + (1.00 − 1.00i)19-s + (−0.605 − 0.605i)21-s − 8.15i·23-s + 1.00i·25-s + (3.18 + 3.18i)27-s + (0.122 − 0.122i)29-s + 2.46·31-s + ⋯
L(s)  = 1  + (0.349 − 0.349i)3-s + (−0.316 − 0.316i)5-s − 0.377i·7-s + 0.755i·9-s + (0.244 + 0.244i)11-s + (−0.776 + 0.776i)13-s − 0.221·15-s + 1.13·17-s + (0.230 − 0.230i)19-s + (−0.132 − 0.132i)21-s − 1.70i·23-s + 0.200i·25-s + (0.613 + 0.613i)27-s + (0.0228 − 0.0228i)29-s + 0.442·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.988 + 0.152i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.988 + 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.911098775\)
\(L(\frac12)\) \(\approx\) \(1.911098775\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.605 + 0.605i)T - 3iT^{2} \)
11 \( 1 + (-0.812 - 0.812i)T + 11iT^{2} \)
13 \( 1 + (2.79 - 2.79i)T - 13iT^{2} \)
17 \( 1 - 4.69T + 17T^{2} \)
19 \( 1 + (-1.00 + 1.00i)T - 19iT^{2} \)
23 \( 1 + 8.15iT - 23T^{2} \)
29 \( 1 + (-0.122 + 0.122i)T - 29iT^{2} \)
31 \( 1 - 2.46T + 31T^{2} \)
37 \( 1 + (-3.53 - 3.53i)T + 37iT^{2} \)
41 \( 1 - 4.84iT - 41T^{2} \)
43 \( 1 + (-2.21 - 2.21i)T + 43iT^{2} \)
47 \( 1 - 3.94T + 47T^{2} \)
53 \( 1 + (-7.19 - 7.19i)T + 53iT^{2} \)
59 \( 1 + (-6.54 - 6.54i)T + 59iT^{2} \)
61 \( 1 + (1.81 - 1.81i)T - 61iT^{2} \)
67 \( 1 + (-0.162 + 0.162i)T - 67iT^{2} \)
71 \( 1 + 6.00iT - 71T^{2} \)
73 \( 1 + 7.09iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + (-7.83 + 7.83i)T - 83iT^{2} \)
89 \( 1 + 3.28iT - 89T^{2} \)
97 \( 1 - 0.453T + 97T^{2} \)
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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

−8.897929796794088980961222402836, −8.170008554406025461939340713190, −7.50172131154145747707747061168, −6.92614785622985906991306416294, −5.92114650105049334262024102381, −4.75838595267050311355108433423, −4.38279197990206704025405887668, −3.06744689692649213166558960948, −2.17167238286201866015860986325, −0.955937581064995049263492335429, 0.854982389538600244742011231579, 2.43407801048962136940854296942, 3.40815731100956069085085915351, 3.86106709497594360990031945546, 5.23070581009260480490115887182, 5.75766403235777767552916117585, 6.81797906505816157075975319933, 7.61888430596877319587567170577, 8.227599780211112323763912448955, 9.193056623740944216747080960598

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