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Properties

Label 2-840-840.293-c0-0-4
Degree $2$
Conductor $840$
Sign $-0.229 - 0.973i$
Analytic cond. $0.419214$
Root an. cond. $0.647467$
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  + (0.707 + 0.707i)2-s + (0.923 + 0.382i)3-s + 1.00i·4-s + (−0.923 − 0.382i)5-s + (0.382 + 0.923i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.382 − 0.923i)10-s + (−0.382 + 0.923i)12-s + (0.541 + 0.541i)13-s − 1.00·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + i·18-s − 1.84i·19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.923 + 0.382i)3-s + 1.00i·4-s + (−0.923 − 0.382i)5-s + (0.382 + 0.923i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.382 − 0.923i)10-s + (−0.382 + 0.923i)12-s + (0.541 + 0.541i)13-s − 1.00·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + i·18-s − 1.84i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.525658252\)
\(L(\frac12)\) \(\approx\) \(1.525658252\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + T^{2} \)
13 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 1.84iT - T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 1.84T + T^{2} \)
61 \( 1 - 0.765T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.78585464720617963781201474139, −9.155466524746324999096063365647, −9.000014566274115645308147978887, −8.151749942732306014185869971723, −7.17439059037770125231148667983, −6.48639996496288080123107432743, −5.06682508792269630662216475144, −4.41911498370607135580904484895, −3.39360023381997439146620599089, −2.61627501555961656032148228511, 1.32865907606311989041910774252, 2.98122520361541968056693347761, 3.54891342543607918975105032350, 4.24286165532814368269559004282, 5.78708767623472826715170287234, 6.78267052250107349946304468899, 7.54516678965835123501355622276, 8.440838059413033704745552512358, 9.542679093651995019120629520906, 10.24814765369641652022554755161

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