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Properties

Label 2-1400-35.13-c1-0-19
Degree $2$
Conductor $1400$
Sign $0.999 + 0.0206i$
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  + (0.923 − 0.923i)3-s + (2.48 + 0.915i)7-s + 1.29i·9-s + 1.21·11-s + (0.996 − 0.996i)13-s + (0.567 + 0.567i)17-s − 0.103·19-s + (3.13 − 1.44i)21-s + (1.43 + 1.43i)23-s + (3.96 + 3.96i)27-s + 4.97i·29-s − 4.35i·31-s + (1.12 − 1.12i)33-s + (−2.54 + 2.54i)37-s − 1.83i·39-s + ⋯
L(s)  = 1  + (0.532 − 0.532i)3-s + (0.938 + 0.345i)7-s + 0.431i·9-s + 0.366·11-s + (0.276 − 0.276i)13-s + (0.137 + 0.137i)17-s − 0.0237·19-s + (0.684 − 0.315i)21-s + (0.298 + 0.298i)23-s + (0.763 + 0.763i)27-s + 0.924i·29-s − 0.781i·31-s + (0.195 − 0.195i)33-s + (−0.417 + 0.417i)37-s − 0.294i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.999 + 0.0206i$
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: \((2,\ 1400,\ (\ :1/2),\ 0.999 + 0.0206i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.354325465\)
\(L(\frac12)\) \(\approx\) \(2.354325465\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.48 - 0.915i)T \)
good3 \( 1 + (-0.923 + 0.923i)T - 3iT^{2} \)
11 \( 1 - 1.21T + 11T^{2} \)
13 \( 1 + (-0.996 + 0.996i)T - 13iT^{2} \)
17 \( 1 + (-0.567 - 0.567i)T + 17iT^{2} \)
19 \( 1 + 0.103T + 19T^{2} \)
23 \( 1 + (-1.43 - 1.43i)T + 23iT^{2} \)
29 \( 1 - 4.97iT - 29T^{2} \)
31 \( 1 + 4.35iT - 31T^{2} \)
37 \( 1 + (2.54 - 2.54i)T - 37iT^{2} \)
41 \( 1 + 7.06iT - 41T^{2} \)
43 \( 1 + (-3.11 - 3.11i)T + 43iT^{2} \)
47 \( 1 + (-7.23 - 7.23i)T + 47iT^{2} \)
53 \( 1 + (-0.882 - 0.882i)T + 53iT^{2} \)
59 \( 1 + 8.28T + 59T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 + (7.07 - 7.07i)T - 67iT^{2} \)
71 \( 1 - 0.329T + 71T^{2} \)
73 \( 1 + (-11.3 + 11.3i)T - 73iT^{2} \)
79 \( 1 + 13.6iT - 79T^{2} \)
83 \( 1 + (-4.61 + 4.61i)T - 83iT^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + (2.40 + 2.40i)T + 97iT^{2} \)
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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

−9.248906296538156127146648527852, −8.738529168290621622216152857533, −7.83670827042299396700105752287, −7.45294843183479727215862252735, −6.30801532729708180044084901160, −5.36553135318048531238748369554, −4.54206007811450403078444896184, −3.32420226319807841272320617386, −2.23540048129232389239861714598, −1.32389342409451089672067900574, 1.09679308713651853307698736737, 2.45726480125891447067033410005, 3.67024803078235074771773146215, 4.29896649625542274169146110471, 5.23718903118891781333463049878, 6.34208374165087736116379492669, 7.19418821496050971574837642568, 8.134924395622983323881129263535, 8.799064613122925419333800524362, 9.455938431213634195246369203793

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