Properties

Label 2-1400-35.4-c1-0-8
Degree $2$
Conductor $1400$
Sign $-0.398 - 0.917i$
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  + (2.23 + 1.29i)3-s + (−2.62 − 0.292i)7-s + (1.83 + 3.18i)9-s + (−0.839 + 1.45i)11-s + 4.84i·13-s + (−1.73 − i)17-s + (3.42 + 5.92i)19-s + (−5.50 − 4.05i)21-s + (1.95 − 1.13i)23-s + 1.75i·27-s − 3.32·29-s + (−4.58 + 7.94i)31-s + (−3.75 + 2.16i)33-s + (−2.46 + 1.42i)37-s + (−6.26 + 10.8i)39-s + ⋯
L(s)  = 1  + (1.29 + 0.746i)3-s + (−0.993 − 0.110i)7-s + (0.613 + 1.06i)9-s + (−0.253 + 0.438i)11-s + 1.34i·13-s + (−0.420 − 0.242i)17-s + (0.785 + 1.36i)19-s + (−1.20 − 0.884i)21-s + (0.408 − 0.235i)23-s + 0.337i·27-s − 0.616·29-s + (−0.823 + 1.42i)31-s + (−0.653 + 0.377i)33-s + (−0.405 + 0.233i)37-s + (−1.00 + 1.73i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.398 - 0.917i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.398 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.982367065\)
\(L(\frac12)\) \(\approx\) \(1.982367065\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.292i)T \)
good3 \( 1 + (-2.23 - 1.29i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.839 - 1.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.84iT - 13T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.42 - 5.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.95 + 1.13i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 + (4.58 - 7.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.46 - 1.42i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.52T + 41T^{2} \)
43 \( 1 + 6.58iT - 43T^{2} \)
47 \( 1 + (10.5 - 6.10i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.48 - 3.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.24 - 5.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.98 - 2.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (10.1 + 5.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.84 - 4.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.5iT - 83T^{2} \)
89 \( 1 + (2.92 + 5.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−9.660115579129071054122156216544, −9.134598991071336372985738035907, −8.465667715006334553036033404908, −7.40601941925072011855320194832, −6.78431449328791319889021002431, −5.59401878127341062668863014753, −4.42794918281507476013640154572, −3.71475577305925059501973872634, −2.94245093298279693151184793548, −1.84245998557694801457056244725, 0.65657766405335485413461031699, 2.29607198322205803749143221539, 3.01921987993145446284872454048, 3.70510745010461925890131608610, 5.23984681211333042320427904408, 6.13703774477522444924411746943, 7.15745129007140215897707125373, 7.65374273145414675035779643246, 8.518445091486468726930794297350, 9.198796884938854135501735391389

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