Properties

Label 2-1400-5.4-c1-0-13
Degree $2$
Conductor $1400$
Sign $0.894 + 0.447i$
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  − 2i·3-s + i·7-s − 9-s + 5·11-s + 8i·17-s + 2·19-s + 2·21-s + 7i·23-s − 4i·27-s + 3·29-s + 4·31-s − 10i·33-s i·37-s − 2·41-s − 3i·43-s + ⋯
L(s)  = 1  − 1.15i·3-s + 0.377i·7-s − 0.333·9-s + 1.50·11-s + 1.94i·17-s + 0.458·19-s + 0.436·21-s + 1.45i·23-s − 0.769i·27-s + 0.557·29-s + 0.718·31-s − 1.74i·33-s − 0.164i·37-s − 0.312·41-s − 0.457i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.904987991\)
\(L(\frac12)\) \(\approx\) \(1.904987991\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 + 2iT - 3T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 8iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 7iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 3iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 13iT - 67T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−9.412948877245156641979279347824, −8.539009840711224179290071817799, −7.899487964321490089040926769421, −6.96301986838725771611435179009, −6.34985073221161005623696649485, −5.66404157382360935032578202970, −4.28063470076894864834172222196, −3.38103360156599868723252100317, −1.91263005328456586204856707282, −1.23338929273275849222745481597, 0.970384414703730431761788835925, 2.73022877991022426297831627029, 3.74444079350678939419962713899, 4.52572820501920174969384612198, 5.12586281933614860100732679464, 6.47038276679645969553838859863, 7.02073894909483035862929331929, 8.160754162375358366150802777326, 9.215407947505280473054234483484, 9.476163015046336707780206413887

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