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Properties

Label 2-1400-7.4-c1-0-8
Degree $2$
Conductor $1400$
Sign $0.701 - 0.712i$
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 − 1.73i)3-s + (−0.5 + 2.59i)7-s + (−0.499 − 0.866i)9-s + (−2 + 3.46i)11-s + 2·13-s + (−1.5 + 2.59i)17-s + (4 + 3.46i)21-s + (1.5 + 2.59i)23-s + 4.00·27-s − 6·29-s + (−4.5 + 7.79i)31-s + (3.99 + 6.92i)33-s + (2 − 3.46i)39-s + 5·41-s + 6·43-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (−0.188 + 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.603 + 1.04i)11-s + 0.554·13-s + (−0.363 + 0.630i)17-s + (0.872 + 0.755i)21-s + (0.312 + 0.541i)23-s + 0.769·27-s − 1.11·29-s + (−0.808 + 1.39i)31-s + (0.696 + 1.20i)33-s + (0.320 − 0.554i)39-s + 0.780·41-s + 0.914·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.705147888\)
\(L(\frac12)\) \(\approx\) \(1.705147888\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (4.5 - 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7 + 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.5 + 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (5.5 + 9.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11T + 97T^{2} \)
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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

−9.316402721314430828805157942795, −8.911438316034635997633731928727, −7.88581045624189820291944312690, −7.42881538663985532489250437751, −6.48802365056284018082708748710, −5.65265750840310458565433198325, −4.65881410784362455438298351545, −3.33129490705527477394992038943, −2.31856820978026890140390153273, −1.58717608591610793170014415323, 0.64256761870295660791180051862, 2.53047616064698233314009098492, 3.59879437782135210841045002144, 4.06024491269712619656586931581, 5.13849713972030019075298364589, 6.10610431854181666728485279323, 7.13921398510323406508806801667, 7.963149347942184417149369613609, 8.828492515546851920530589745748, 9.442230302341272876940673159998

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