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Properties

Label 2-1400-35.4-c1-0-2
Degree $2$
Conductor $1400$
Sign $-0.208 - 0.978i$
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  + (−0.866 − 0.5i)3-s + (0.866 + 2.5i)7-s + (−1 − 1.73i)9-s + (−1 + 1.73i)11-s + (3.46 + 2i)17-s + (−1 − 1.73i)19-s + (0.500 − 2.59i)21-s + (0.866 − 0.5i)23-s + 5i·27-s − 9·29-s + (−2 + 3.46i)31-s + (1.73 − 0.999i)33-s + (−3.46 + 2i)37-s + 41-s + 9i·43-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.327 + 0.944i)7-s + (−0.333 − 0.577i)9-s + (−0.301 + 0.522i)11-s + (0.840 + 0.485i)17-s + (−0.229 − 0.397i)19-s + (0.109 − 0.566i)21-s + (0.180 − 0.104i)23-s + 0.962i·27-s − 1.67·29-s + (−0.359 + 0.622i)31-s + (0.301 − 0.174i)33-s + (−0.569 + 0.328i)37-s + 0.156·41-s + 1.37i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.208 - 0.978i$
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: \((2,\ 1400,\ (\ :1/2),\ -0.208 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8438836370\)
\(L(\frac12)\) \(\approx\) \(0.8438836370\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 2.5i)T \)
good3 \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - T + 41T^{2} \)
43 \( 1 - 9iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.66 - 5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 + 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + (10.3 + 6i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11iT - 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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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

−9.638996396282050916486415693151, −9.048138778877038820856969586805, −8.183557032470932569577040534226, −7.33485504776220586487453957317, −6.41358048075850871332563902085, −5.63949061459481675727639200303, −5.02516477771015952698130739756, −3.73283270797228823878172899156, −2.62835528059228636525604196630, −1.41421674611134668641749669216, 0.37116713737588800641613554818, 1.94066432002736558143614523660, 3.35922544540925577761682698651, 4.22494646622992729183394093757, 5.31430870361603401102157181646, 5.73011305717528525436359989637, 7.02291128186673319704000025023, 7.69486275974375710735575818510, 8.412490122837720619156473862869, 9.483535054426076803971367886924

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