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Properties

Label 2-840-35.13-c1-0-13
Degree $2$
Conductor $840$
Sign $0.950 + 0.312i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
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  + (0.707 − 0.707i)3-s + (1.98 − 1.02i)5-s + (−0.630 + 2.56i)7-s − 1.00i·9-s + 2.36·11-s + (0.918 − 0.918i)13-s + (0.679 − 2.13i)15-s + (3.62 + 3.62i)17-s − 1.07·19-s + (1.37 + 2.26i)21-s + (1.45 + 1.45i)23-s + (2.89 − 4.07i)25-s + (−0.707 − 0.707i)27-s − 7.72i·29-s + 1.21i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.888 − 0.458i)5-s + (−0.238 + 0.971i)7-s − 0.333i·9-s + 0.712·11-s + (0.254 − 0.254i)13-s + (0.175 − 0.550i)15-s + (0.878 + 0.878i)17-s − 0.247·19-s + (0.299 + 0.493i)21-s + (0.302 + 0.302i)23-s + (0.578 − 0.815i)25-s + (−0.136 − 0.136i)27-s − 1.43i·29-s + 0.218i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.950 + 0.312i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.950 + 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13663 - 0.341856i\)
\(L(\frac12)\) \(\approx\) \(2.13663 - 0.341856i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-1.98 + 1.02i)T \)
7 \( 1 + (0.630 - 2.56i)T \)
good11 \( 1 - 2.36T + 11T^{2} \)
13 \( 1 + (-0.918 + 0.918i)T - 13iT^{2} \)
17 \( 1 + (-3.62 - 3.62i)T + 17iT^{2} \)
19 \( 1 + 1.07T + 19T^{2} \)
23 \( 1 + (-1.45 - 1.45i)T + 23iT^{2} \)
29 \( 1 + 7.72iT - 29T^{2} \)
31 \( 1 - 1.21iT - 31T^{2} \)
37 \( 1 + (-2.38 + 2.38i)T - 37iT^{2} \)
41 \( 1 - 8.42iT - 41T^{2} \)
43 \( 1 + (-0.879 - 0.879i)T + 43iT^{2} \)
47 \( 1 + (-3.67 - 3.67i)T + 47iT^{2} \)
53 \( 1 + (7.88 + 7.88i)T + 53iT^{2} \)
59 \( 1 + 8.24T + 59T^{2} \)
61 \( 1 + 10.1iT - 61T^{2} \)
67 \( 1 + (-6.62 + 6.62i)T - 67iT^{2} \)
71 \( 1 + 8.21T + 71T^{2} \)
73 \( 1 + (-1.79 + 1.79i)T - 73iT^{2} \)
79 \( 1 - 12.2iT - 79T^{2} \)
83 \( 1 + (0.387 - 0.387i)T - 83iT^{2} \)
89 \( 1 - 7.46T + 89T^{2} \)
97 \( 1 + (-6.92 - 6.92i)T + 97iT^{2} \)
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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

−9.783648710060294793701033297937, −9.399704794375714113342454891677, −8.489060090520632830761795406570, −7.83832276349729648292525489189, −6.33652719737051296738013553363, −6.06276562335619884799187791308, −4.93309779228078351220148586820, −3.57034213982824193455350251803, −2.40672739397633495800306535600, −1.34457061324062197801546404076, 1.35622848261299090535080375660, 2.83758104707817474032115819123, 3.73618445468180378755525227882, 4.81737124050397666793538642565, 5.92325834667284731401905038650, 6.89065545518514908386467495292, 7.50668623746794497388361188811, 8.866647888230659954240949462457, 9.392344885106438594074620188960, 10.29695275186277330995893403910

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