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Properties

Label 2-1400-5.4-c1-0-18
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  + i·3-s + i·7-s + 2·9-s − 11-s − 6i·13-s − 7i·17-s − 19-s − 21-s − 8i·23-s + 5i·27-s + 6·29-s + 4·31-s − i·33-s + 8i·37-s + 6·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.377i·7-s + 0.666·9-s − 0.301·11-s − 1.66i·13-s − 1.69i·17-s − 0.229·19-s − 0.218·21-s − 1.66i·23-s + 0.962i·27-s + 1.11·29-s + 0.718·31-s − 0.174i·33-s + 1.31i·37-s + 0.960·39-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

\(L(1)\) \(\approx\) \(1.635165858\)
\(L(\frac12)\) \(\approx\) \(1.635165858\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 3T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 7iT - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + 15iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 + iT - 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−9.745358688199461289818865521463, −8.646280040694369929067538312951, −8.044388465311646721934293537643, −7.05525234841462589816395295072, −6.20906592445487140484728905006, −4.98777396072363524285581432063, −4.72547677431999923314001115728, −3.27803713921887626994531108069, −2.55468970728621955049122464163, −0.73397297778883513839840362463, 1.33234153155859932189094019486, 2.15448702032286839235693112090, 3.76745244395628661583181959798, 4.34382941248300225089475117596, 5.58025139286356655472514970414, 6.60169324507038376327337793672, 7.04401845026294622597662082657, 7.981934832072708442320858249336, 8.700113316836445840078054960900, 9.721398022295779136169620725401

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