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Properties

Label 2-1680-35.13-c1-0-31
Degree $2$
Conductor $1680$
Sign $0.337 + 0.941i$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
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 + (−1.28 + 1.82i)5-s + (1.97 − 1.75i)7-s − 1.00i·9-s + 2.67·11-s + (1.22 − 1.22i)13-s + (−0.379 − 2.20i)15-s + (−4.74 − 4.74i)17-s − 6.01·19-s + (−0.152 + 2.64i)21-s + (0.175 + 0.175i)23-s + (−1.67 − 4.71i)25-s + (0.707 + 0.707i)27-s + 0.304i·29-s − 7.25i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.576 + 0.816i)5-s + (0.746 − 0.665i)7-s − 0.333i·9-s + 0.805·11-s + (0.340 − 0.340i)13-s + (−0.0979 − 0.568i)15-s + (−1.15 − 1.15i)17-s − 1.38·19-s + (−0.0332 + 0.576i)21-s + (0.0366 + 0.0366i)23-s + (−0.334 − 0.942i)25-s + (0.136 + 0.136i)27-s + 0.0566i·29-s − 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.337 + 0.941i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.337 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9666160054\)
\(L(\frac12)\) \(\approx\) \(0.9666160054\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 - 1.82i)T \)
7 \( 1 + (-1.97 + 1.75i)T \)
good11 \( 1 - 2.67T + 11T^{2} \)
13 \( 1 + (-1.22 + 1.22i)T - 13iT^{2} \)
17 \( 1 + (4.74 + 4.74i)T + 17iT^{2} \)
19 \( 1 + 6.01T + 19T^{2} \)
23 \( 1 + (-0.175 - 0.175i)T + 23iT^{2} \)
29 \( 1 - 0.304iT - 29T^{2} \)
31 \( 1 + 7.25iT - 31T^{2} \)
37 \( 1 + (0.735 - 0.735i)T - 37iT^{2} \)
41 \( 1 - 7.05iT - 41T^{2} \)
43 \( 1 + (0.304 + 0.304i)T + 43iT^{2} \)
47 \( 1 + (0.556 + 0.556i)T + 47iT^{2} \)
53 \( 1 + (4.99 + 4.99i)T + 53iT^{2} \)
59 \( 1 - 7.98T + 59T^{2} \)
61 \( 1 + 5.53iT - 61T^{2} \)
67 \( 1 + (-3.43 + 3.43i)T - 67iT^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + (-10.0 + 10.0i)T - 73iT^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + (-4.88 + 4.88i)T - 83iT^{2} \)
89 \( 1 - 6.91T + 89T^{2} \)
97 \( 1 + (8.84 + 8.84i)T + 97iT^{2} \)
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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

−9.227554134856075831166729035812, −8.359682124720852467783481731572, −7.56381052807990411550264730015, −6.71605393787149457075562495081, −6.18628010824627189373319337202, −4.79015122453914910203636962996, −4.26760075822389304637579975668, −3.39010357787164701224372031453, −2.08790837586906422541795255303, −0.40897104025860906953761123947, 1.31898632942659540312711397691, 2.16869179092582434265487492672, 3.90745183790656071065693580436, 4.48837122708858347288794827588, 5.43501710184490978535617486964, 6.31194169688315421125589296316, 7.02078363115518454260095198069, 8.188046839953138689765655121983, 8.639942541474616738059000796532, 9.118097322703415710389716742651

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