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Properties

Label 2-1400-35.4-c1-0-28
Degree $2$
Conductor $1400$
Sign $-0.211 - 0.977i$
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  + (−2.87 − 1.65i)3-s + (−2.49 + 0.873i)7-s + (3.99 + 6.92i)9-s + (2.12 − 3.67i)11-s − 5.10i·13-s + (0.765 + 0.441i)17-s + (−3.27 − 5.66i)19-s + (8.61 + 1.63i)21-s + (−4.62 + 2.66i)23-s − 16.5i·27-s − 4.54·29-s + (−0.338 + 0.586i)31-s + (−12.1 + 7.03i)33-s + (1.31 − 0.760i)37-s + (−8.46 + 14.6i)39-s + ⋯
L(s)  = 1  + (−1.65 − 0.957i)3-s + (−0.943 + 0.330i)7-s + (1.33 + 2.30i)9-s + (0.640 − 1.10i)11-s − 1.41i·13-s + (0.185 + 0.107i)17-s + (−0.750 − 1.29i)19-s + (1.88 + 0.355i)21-s + (−0.963 + 0.556i)23-s − 3.18i·27-s − 0.843·29-s + (−0.0608 + 0.105i)31-s + (−2.12 + 1.22i)33-s + (0.216 − 0.125i)37-s + (−1.35 + 2.34i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.211 - 0.977i$
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: \((2,\ 1400,\ (\ :1/2),\ -0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.005234221608\)
\(L(\frac12)\) \(\approx\) \(0.005234221608\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.49 - 0.873i)T \)
good3 \( 1 + (2.87 + 1.65i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.10iT - 13T^{2} \)
17 \( 1 + (-0.765 - 0.441i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.27 + 5.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.62 - 2.66i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.54T + 29T^{2} \)
31 \( 1 + (0.338 - 0.586i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.31 + 0.760i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.92T + 41T^{2} \)
43 \( 1 - 5.49iT - 43T^{2} \)
47 \( 1 + (-6.29 + 3.63i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.22 - 1.28i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.97 - 6.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.50 - 7.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.63 + 1.52i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.39T + 71T^{2} \)
73 \( 1 + (-3.34 - 1.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.352 - 0.610i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 + (4.20 + 7.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.614iT - 97T^{2} \)
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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

−8.950147999141594578182587801982, −7.978320695538280267335023906511, −7.12058277564749181949791420220, −6.33729888549796860370680398504, −5.83035359958084870568761634213, −5.20735988903614946749863493383, −3.82801777431257964909096711045, −2.52990837554016864034076681868, −1.00752927017834083792597725115, −0.00325866467249271290085575879, 1.74942880849220527982289302285, 3.92469177246580227228661530770, 4.04190708822745982746786629402, 5.10887110160012503617522675233, 6.20058419196407695772637276086, 6.51146321257483886011532242335, 7.35418341489656274675071463264, 8.920355428093328342130503648347, 9.767926519586038907441412238930, 9.980163405569402674696626050033

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