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Properties

Label 2-1400-35.27-c1-0-18
Degree $2$
Conductor $1400$
Sign $0.296 - 0.955i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
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  + (1.66 + 1.66i)3-s + (2.27 + 1.34i)7-s + 2.56i·9-s + 1.56·11-s + (2.60 + 2.60i)13-s + (2.60 − 2.60i)17-s − 2.64·19-s + (1.56 + 6.04i)21-s + (−0.794 + 0.794i)23-s + (0.731 − 0.731i)27-s + 6.68i·29-s − 9.43i·31-s + (2.60 + 2.60i)33-s + (−2.82 − 2.82i)37-s + 8.68i·39-s + ⋯
L(s)  = 1  + (0.962 + 0.962i)3-s + (0.861 + 0.507i)7-s + 0.853i·9-s + 0.470·11-s + (0.722 + 0.722i)13-s + (0.631 − 0.631i)17-s − 0.607·19-s + (0.340 + 1.31i)21-s + (−0.165 + 0.165i)23-s + (0.140 − 0.140i)27-s + 1.24i·29-s − 1.69i·31-s + (0.453 + 0.453i)33-s + (−0.464 − 0.464i)37-s + 1.39i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.296 - 0.955i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.296 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.715162825\)
\(L(\frac12)\) \(\approx\) \(2.715162825\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.27 - 1.34i)T \)
good3 \( 1 + (-1.66 - 1.66i)T + 3iT^{2} \)
11 \( 1 - 1.56T + 11T^{2} \)
13 \( 1 + (-2.60 - 2.60i)T + 13iT^{2} \)
17 \( 1 + (-2.60 + 2.60i)T - 17iT^{2} \)
19 \( 1 + 2.64T + 19T^{2} \)
23 \( 1 + (0.794 - 0.794i)T - 23iT^{2} \)
29 \( 1 - 6.68iT - 29T^{2} \)
31 \( 1 + 9.43iT - 31T^{2} \)
37 \( 1 + (2.82 + 2.82i)T + 37iT^{2} \)
41 \( 1 - 2.64iT - 41T^{2} \)
43 \( 1 + (6.45 - 6.45i)T - 43iT^{2} \)
47 \( 1 + (-1.66 + 1.66i)T - 47iT^{2} \)
53 \( 1 + (-7.24 + 7.24i)T - 53iT^{2} \)
59 \( 1 + 12.0T + 59T^{2} \)
61 \( 1 - 9.43iT - 61T^{2} \)
67 \( 1 + (-3.62 - 3.62i)T + 67iT^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 + (6.67 + 6.67i)T + 73iT^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 + (9.47 + 9.47i)T + 83iT^{2} \)
89 \( 1 - 9.43T + 89T^{2} \)
97 \( 1 + (5.52 - 5.52i)T - 97iT^{2} \)
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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

−9.543933443981995960140458355785, −8.913832909539760183284091876926, −8.413514683412712572941438486978, −7.53986605987736425295188604915, −6.41961675211597587866568203904, −5.38667999701862761714694481578, −4.44748557832365193096283561931, −3.77725205314627194403071516401, −2.74307097029771565760046149994, −1.60142988333774182713016272575, 1.13910070527970706812104428059, 1.95806496177385972264585853452, 3.19425644083610355949242626304, 4.06065836267045726538688970266, 5.24751090484326628025756261650, 6.33935541742697661788749717993, 7.11350461177578592714652341705, 8.020172123728555761574065875363, 8.305036299934197856785063366379, 9.083499350847383561467002658323

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