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Properties

Label 2-1400-1400.1021-c0-0-0
Degree $2$
Conductor $1400$
Sign $-0.535 - 0.844i$
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  + (−0.309 + 0.951i)2-s + (−0.951 − 0.690i)3-s + (−0.809 − 0.587i)4-s + (−0.951 − 0.309i)5-s + (0.951 − 0.690i)6-s − 7-s + (0.809 − 0.587i)8-s + (0.118 + 0.363i)9-s + (0.587 − 0.809i)10-s + (0.363 + 1.11i)12-s + (−0.363 − 1.11i)13-s + (0.309 − 0.951i)14-s + (0.690 + 0.951i)15-s + (0.309 + 0.951i)16-s − 0.381·18-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.951 − 0.690i)3-s + (−0.809 − 0.587i)4-s + (−0.951 − 0.309i)5-s + (0.951 − 0.690i)6-s − 7-s + (0.809 − 0.587i)8-s + (0.118 + 0.363i)9-s + (0.587 − 0.809i)10-s + (0.363 + 1.11i)12-s + (−0.363 − 1.11i)13-s + (0.309 − 0.951i)14-s + (0.690 + 0.951i)15-s + (0.309 + 0.951i)16-s − 0.381·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.535 - 0.844i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ -0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1641909623\)
\(L(\frac12)\) \(\approx\) \(0.1641909623\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + T \)
good3 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−9.929853087427870369365497987937, −9.092945875031152543799271022875, −8.195520642281225773788377613018, −7.32469654630836265254292226433, −7.01796097134058085074293548121, −5.81426637821549381492400459473, −5.56431089845192899937052131260, −4.29908048168071183420264808618, −3.25653215320504421441930454776, −1.04958346167430216007517007516, 0.21081003325642543076509382466, 2.35155292541466349688018106358, 3.48607617149681610483187808566, 4.28461936298207826056382780135, 4.89403118273409426741200302999, 6.21809941591680084889248442738, 7.00804181679847158553252432590, 8.071817211591496301052886011327, 8.920161625485446722296699071497, 9.818296457745604142372421733040

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