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Properties

Label 2-2240-16.5-c1-0-45
Degree $2$
Conductor $2240$
Sign $-0.976 + 0.215i$
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  + (1.35 − 1.35i)3-s + (−0.707 − 0.707i)5-s − i·7-s − 0.692i·9-s + (−2.35 − 2.35i)11-s + (−0.605 + 0.605i)13-s − 1.92·15-s − 1.39·17-s + (2.68 − 2.68i)19-s + (−1.35 − 1.35i)21-s − 5.65i·23-s + 1.00i·25-s + (3.13 + 3.13i)27-s + (−0.938 + 0.938i)29-s − 8.67·31-s + ⋯
L(s)  = 1  + (0.784 − 0.784i)3-s + (−0.316 − 0.316i)5-s − 0.377i·7-s − 0.230i·9-s + (−0.709 − 0.709i)11-s + (−0.168 + 0.168i)13-s − 0.496·15-s − 0.339·17-s + (0.616 − 0.616i)19-s + (−0.296 − 0.296i)21-s − 1.17i·23-s + 0.200i·25-s + (0.603 + 0.603i)27-s + (−0.174 + 0.174i)29-s − 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.976 + 0.215i$
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: \((2,\ 2240,\ (\ :1/2),\ -0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.256458114\)
\(L(\frac12)\) \(\approx\) \(1.256458114\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 + 1.35i)T - 3iT^{2} \)
11 \( 1 + (2.35 + 2.35i)T + 11iT^{2} \)
13 \( 1 + (0.605 - 0.605i)T - 13iT^{2} \)
17 \( 1 + 1.39T + 17T^{2} \)
19 \( 1 + (-2.68 + 2.68i)T - 19iT^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 + (0.938 - 0.938i)T - 29iT^{2} \)
31 \( 1 + 8.67T + 31T^{2} \)
37 \( 1 + (1.49 + 1.49i)T + 37iT^{2} \)
41 \( 1 + 6.63iT - 41T^{2} \)
43 \( 1 + (-2.62 - 2.62i)T + 43iT^{2} \)
47 \( 1 + 7.68T + 47T^{2} \)
53 \( 1 + (6.01 + 6.01i)T + 53iT^{2} \)
59 \( 1 + (4.55 + 4.55i)T + 59iT^{2} \)
61 \( 1 + (-1.47 + 1.47i)T - 61iT^{2} \)
67 \( 1 + (4.96 - 4.96i)T - 67iT^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + 0.758iT - 73T^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 + (-8.55 + 8.55i)T - 83iT^{2} \)
89 \( 1 + 7.52iT - 89T^{2} \)
97 \( 1 - 0.0599T + 97T^{2} \)
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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

−8.648981641885700365731259211566, −7.84450159922328750863164855883, −7.33560686380041937114487450623, −6.58475779314820540079445326030, −5.44734020622120098031892355484, −4.66597158941220420021024247902, −3.53756621268922548151239688684, −2.71137639869978613782273135110, −1.72154912471453224789479259835, −0.36480668784479099637664358363, 1.80909508623494592595467248400, 2.94008059592813151232787260885, 3.53304579238311565724092670628, 4.48509927760501288796208752940, 5.29540819033956195170881840430, 6.22677444952416964582170471607, 7.39455099489563595646697511928, 7.80234964992525426482682587469, 8.744800261391813308003894499387, 9.461545073859023443632295450407

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