Properties

Label 2-1400-56.13-c0-0-1
Degree $2$
Conductor $1400$
Sign $1$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 7-s + 0.999·8-s − 9-s + 1.73i·11-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + (1.49 − 0.866i)22-s + 23-s + (−0.499 + 0.866i)28-s + 1.73i·29-s + (−0.499 + 0.866i)32-s + (0.499 − 0.866i)36-s + 1.73i·37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 7-s + 0.999·8-s − 9-s + 1.73i·11-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + (1.49 − 0.866i)22-s + 23-s + (−0.499 + 0.866i)28-s + 1.73i·29-s + (−0.499 + 0.866i)32-s + (0.499 − 0.866i)36-s + 1.73i·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8170917516\)
\(L(\frac12)\) \(\approx\) \(0.8170917516\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 - T \)
good3 \( 1 + T^{2} \)
11 \( 1 - 1.73iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.779024569338101610645830766425, −8.974190698158375424781899021951, −8.374357162294601641240779888785, −7.51506998379007194065014130210, −6.80216194492556374663850481879, −5.14470689008525013489662826487, −4.77125087247963427399857683850, −3.52479085271814207905637985186, −2.43350354613194614869655619725, −1.49688086274837542164910117526, 0.890299391740152946017137073877, 2.53036363345507421042197738697, 3.90274245315059242989351925157, 5.06511350778806645121820480257, 5.74418343413207275652897321280, 6.36151742882763868532942070243, 7.59126366490503593951134228904, 8.175960017039092695013840151809, 8.752864014611837125206330724510, 9.420855792220089780829821102286

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