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Properties

Label 2-2240-56.27-c1-0-15
Degree $2$
Conductor $2240$
Sign $-0.964 - 0.265i$
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  + 2.59i·3-s + 5-s + (2.28 + 1.33i)7-s − 3.73·9-s + 3.05·11-s − 6.11·13-s + 2.59i·15-s + 1.25i·17-s + 3.37i·19-s + (−3.47 + 5.92i)21-s − 4.24i·23-s + 25-s − 1.92i·27-s + 1.52i·29-s − 9.46·31-s + ⋯
L(s)  = 1  + 1.49i·3-s + 0.447·5-s + (0.862 + 0.505i)7-s − 1.24·9-s + 0.922·11-s − 1.69·13-s + 0.670i·15-s + 0.304i·17-s + 0.774i·19-s + (−0.757 + 1.29i)21-s − 0.884i·23-s + 0.200·25-s − 0.369i·27-s + 0.283i·29-s − 1.70·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.665830595\)
\(L(\frac12)\) \(\approx\) \(1.665830595\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (-2.28 - 1.33i)T \)
good3 \( 1 - 2.59iT - 3T^{2} \)
11 \( 1 - 3.05T + 11T^{2} \)
13 \( 1 + 6.11T + 13T^{2} \)
17 \( 1 - 1.25iT - 17T^{2} \)
19 \( 1 - 3.37iT - 19T^{2} \)
23 \( 1 + 4.24iT - 23T^{2} \)
29 \( 1 - 1.52iT - 29T^{2} \)
31 \( 1 + 9.46T + 31T^{2} \)
37 \( 1 - 3.12iT - 37T^{2} \)
41 \( 1 - 3.19iT - 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 - 7.89T + 47T^{2} \)
53 \( 1 - 12.5iT - 53T^{2} \)
59 \( 1 - 14.5iT - 59T^{2} \)
61 \( 1 - 0.274T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 + 7.23iT - 71T^{2} \)
73 \( 1 + 0.742iT - 73T^{2} \)
79 \( 1 + 10.1iT - 79T^{2} \)
83 \( 1 - 6.15iT - 83T^{2} \)
89 \( 1 + 5.78iT - 89T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−9.216685080529609237613708844418, −9.051693949180047130811393647604, −8.017155566582138572069483576499, −7.08879676476291809738347630481, −5.98036678221263534483362416480, −5.22457718088901409865211619481, −4.63342374938834606424198990338, −3.88788983678041945039597847417, −2.77050422442680655676513387530, −1.70625371121351427592354242088, 0.54762498381970602934108927591, 1.75627920232348742475730651179, 2.26679147801741179816153098034, 3.66278611501712773224936042185, 4.90491436678531318925891736009, 5.51505905862632104547523866284, 6.72267033405188311951438988045, 7.10813209180893317425061025727, 7.65824711202530936217811334305, 8.524802414789225119915064484503

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