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Properties

Label 2-1400-35.9-c1-0-21
Degree $2$
Conductor $1400$
Sign $0.999 + 0.0121i$
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  + (2.19 − 1.26i)3-s + (−2.31 + 1.28i)7-s + (1.70 − 2.95i)9-s + (2.11 + 3.66i)11-s + 5.98i·13-s + (6.61 − 3.81i)17-s + (3.47 − 6.01i)19-s + (−3.43 + 5.74i)21-s + (1.48 + 0.858i)23-s − 1.04i·27-s − 5.18·29-s + (0.254 + 0.441i)31-s + (9.26 + 5.35i)33-s + (3.45 + 1.99i)37-s + (7.57 + 13.1i)39-s + ⋯
L(s)  = 1  + (1.26 − 0.730i)3-s + (−0.873 + 0.486i)7-s + (0.568 − 0.984i)9-s + (0.637 + 1.10i)11-s + 1.66i·13-s + (1.60 − 0.926i)17-s + (0.796 − 1.37i)19-s + (−0.750 + 1.25i)21-s + (0.310 + 0.178i)23-s − 0.200i·27-s − 0.962·29-s + (0.0457 + 0.0792i)31-s + (1.61 + 0.931i)33-s + (0.567 + 0.327i)37-s + (1.21 + 2.10i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.999 + 0.0121i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.999 + 0.0121i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.570450415\)
\(L(\frac12)\) \(\approx\) \(2.570450415\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.31 - 1.28i)T \)
good3 \( 1 + (-2.19 + 1.26i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.11 - 3.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.98iT - 13T^{2} \)
17 \( 1 + (-6.61 + 3.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.47 + 6.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.48 - 0.858i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.18T + 29T^{2} \)
31 \( 1 + (-0.254 - 0.441i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.45 - 1.99i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.71T + 41T^{2} \)
43 \( 1 - 4.17iT - 43T^{2} \)
47 \( 1 + (-1.64 - 0.946i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.53 + 3.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.79 + 4.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.15 + 3.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.4 - 6.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.08T + 71T^{2} \)
73 \( 1 + (4.66 - 2.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.673 - 1.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.70iT - 83T^{2} \)
89 \( 1 + (-5.42 + 9.39i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.370832812598081628303067407602, −9.043333166311241968144660703547, −7.83674941842343573471780585942, −7.07164304573753356130723629531, −6.76851310803983340591151566546, −5.42640417377732980468790404982, −4.25967061432341849132821540436, −3.19425618646936668693410345746, −2.45559456837015036332936229284, −1.37152031369674969522702897856, 1.05883929694328166404571344149, 2.88461905711385455488082261572, 3.52401117291107637748418479893, 3.88751326596246113497659353579, 5.58005754730368061442617888598, 6.03019908387211388438712396630, 7.59455288234191284613494121488, 7.930308975382482277476641776724, 8.845059495188935727219993185401, 9.542570335424757487638481777198

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