Properties

Label 2-1400-35.13-c1-0-2
Degree $2$
Conductor $1400$
Sign $-0.768 + 0.640i$
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  + (−1.03 + 1.03i)3-s + (−1.58 + 2.11i)7-s + 0.841i·9-s − 2.34·11-s + (−1.96 + 1.96i)13-s + (5.15 + 5.15i)17-s + 3.74·19-s + (−0.547 − 3.84i)21-s + (−6.08 − 6.08i)23-s + (−3.99 − 3.99i)27-s − 5.89i·29-s + 1.56i·31-s + (2.43 − 2.43i)33-s + (−1.53 + 1.53i)37-s − 4.08i·39-s + ⋯
L(s)  = 1  + (−0.599 + 0.599i)3-s + (−0.600 + 0.799i)7-s + 0.280i·9-s − 0.707·11-s + (−0.544 + 0.544i)13-s + (1.25 + 1.25i)17-s + 0.859·19-s + (−0.119 − 0.839i)21-s + (−1.26 − 1.26i)23-s + (−0.768 − 0.768i)27-s − 1.09i·29-s + 0.281i·31-s + (0.424 − 0.424i)33-s + (−0.252 + 0.252i)37-s − 0.653i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.768 + 0.640i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.768 + 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2831294642\)
\(L(\frac12)\) \(\approx\) \(0.2831294642\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 2.11i)T \)
good3 \( 1 + (1.03 - 1.03i)T - 3iT^{2} \)
11 \( 1 + 2.34T + 11T^{2} \)
13 \( 1 + (1.96 - 1.96i)T - 13iT^{2} \)
17 \( 1 + (-5.15 - 5.15i)T + 17iT^{2} \)
19 \( 1 - 3.74T + 19T^{2} \)
23 \( 1 + (6.08 + 6.08i)T + 23iT^{2} \)
29 \( 1 + 5.89iT - 29T^{2} \)
31 \( 1 - 1.56iT - 31T^{2} \)
37 \( 1 + (1.53 - 1.53i)T - 37iT^{2} \)
41 \( 1 - 9.51iT - 41T^{2} \)
43 \( 1 + (1.86 + 1.86i)T + 43iT^{2} \)
47 \( 1 + (4.59 + 4.59i)T + 47iT^{2} \)
53 \( 1 + (3.88 + 3.88i)T + 53iT^{2} \)
59 \( 1 + 4.62T + 59T^{2} \)
61 \( 1 - 2.00iT - 61T^{2} \)
67 \( 1 + (2.69 - 2.69i)T - 67iT^{2} \)
71 \( 1 + 0.392T + 71T^{2} \)
73 \( 1 + (-7.08 + 7.08i)T - 73iT^{2} \)
79 \( 1 - 4.98iT - 79T^{2} \)
83 \( 1 + (-9.36 + 9.36i)T - 83iT^{2} \)
89 \( 1 + 9.83T + 89T^{2} \)
97 \( 1 + (11.8 + 11.8i)T + 97iT^{2} \)
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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

−10.00760702215756282695957283502, −9.600073617857281028580482968631, −8.299357819844271520288461925091, −7.86804824345882465571688827511, −6.53812059240906519353257681196, −5.83024466436457361552277723568, −5.14767650695512011232279835896, −4.23666741170014597144807090985, −3.09225320382537954649640098255, −1.99393978665417869696290281825, 0.12943984380933138356408794147, 1.27610163693710783514492133731, 2.95099976764829053526581197881, 3.70502808557199996093235223622, 5.18369614407522922523518414465, 5.65449958229838096279056307274, 6.73209575923739456072781117106, 7.49601146301025600219629417097, 7.81091207586197371438204393618, 9.413089638705375358208893809633

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