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Properties

Label 2-1400-35.4-c1-0-23
Degree $2$
Conductor $1400$
Sign $-0.669 + 0.742i$
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  + (−1.09 − 0.632i)3-s + (2.16 + 1.51i)7-s + (−0.699 − 1.21i)9-s + (−3.21 + 5.57i)11-s − 1.10i·13-s + (−3.89 − 2.25i)17-s + (−2.93 − 5.08i)19-s + (−1.41 − 3.03i)21-s + (7.59 − 4.38i)23-s + 5.56i·27-s − 3.87·29-s + (3.33 − 5.77i)31-s + (7.04 − 4.06i)33-s + (1.23 − 0.713i)37-s + (−0.698 + 1.21i)39-s + ⋯
L(s)  = 1  + (−0.632 − 0.365i)3-s + (0.819 + 0.573i)7-s + (−0.233 − 0.403i)9-s + (−0.969 + 1.67i)11-s − 0.306i·13-s + (−0.945 − 0.546i)17-s + (−0.673 − 1.16i)19-s + (−0.308 − 0.662i)21-s + (1.58 − 0.914i)23-s + 1.07i·27-s − 0.719·29-s + (0.598 − 1.03i)31-s + (1.22 − 0.708i)33-s + (0.203 − 0.117i)37-s + (−0.111 + 0.193i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.669 + 0.742i$
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: \((2,\ 1400,\ (\ :1/2),\ -0.669 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6067472005\)
\(L(\frac12)\) \(\approx\) \(0.6067472005\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 - 1.51i)T \)
good3 \( 1 + (1.09 + 0.632i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (3.21 - 5.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.10iT - 13T^{2} \)
17 \( 1 + (3.89 + 2.25i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.93 + 5.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.59 + 4.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 + (-3.33 + 5.77i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.23 + 0.713i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 + 6.46iT - 43T^{2} \)
47 \( 1 + (-7.32 + 4.23i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.665 + 0.384i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.898 - 1.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.51 + 2.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.47 + 1.42i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 + (11.6 + 6.69i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.65 + 8.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.20iT - 83T^{2} \)
89 \( 1 + (1.46 + 2.53i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.96iT - 97T^{2} \)
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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

−9.073831404928577226748468523449, −8.631146640703950355871732168184, −7.35026386287405469303065071220, −6.99674448623504453399052583410, −5.92455571595191650938877027895, −4.93251426878323712718807054459, −4.59336525213272969565699740654, −2.78998479457172555656813238590, −1.97609397618878467807583797758, −0.26633375313647875383416056101, 1.40855354836451163076067480915, 2.87170423137407037957607938275, 4.00075125523885822703997940850, 4.96376250141962449618484440930, 5.57974604672004484380874308685, 6.42580896750375729529045414354, 7.55386132288612833244761879415, 8.330023479523750690299371999568, 8.823108129586617115335175088325, 10.22942315156860788431695044235

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