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Properties

Label 2-1260-12.11-c1-0-23
Degree $2$
Conductor $1260$
Sign $0.981 + 0.191i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (−1.31 + 0.526i)2-s + (1.44 − 1.38i)4-s − i·5-s − i·7-s + (−1.17 + 2.57i)8-s + (0.526 + 1.31i)10-s + 5.94·11-s + 4.86·13-s + (0.526 + 1.31i)14-s + (0.182 − 3.99i)16-s + 6.18i·17-s − 6.20i·19-s + (−1.38 − 1.44i)20-s + (−7.79 + 3.12i)22-s − 7.82·23-s + ⋯
L(s)  = 1  + (−0.928 + 0.372i)2-s + (0.723 − 0.690i)4-s − 0.447i·5-s − 0.377i·7-s + (−0.414 + 0.910i)8-s + (0.166 + 0.415i)10-s + 1.79·11-s + 1.34·13-s + (0.140 + 0.350i)14-s + (0.0456 − 0.998i)16-s + 1.50i·17-s − 1.42i·19-s + (−0.308 − 0.323i)20-s + (−1.66 + 0.666i)22-s − 1.63·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.981 + 0.191i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.981 + 0.191i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.254178906\)
\(L(\frac12)\) \(\approx\) \(1.254178906\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.526i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + iT \)
good11 \( 1 - 5.94T + 11T^{2} \)
13 \( 1 - 4.86T + 13T^{2} \)
17 \( 1 - 6.18iT - 17T^{2} \)
19 \( 1 + 6.20iT - 19T^{2} \)
23 \( 1 + 7.82T + 23T^{2} \)
29 \( 1 - 1.48iT - 29T^{2} \)
31 \( 1 - 3.22iT - 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + 0.728iT - 41T^{2} \)
43 \( 1 - 0.297iT - 43T^{2} \)
47 \( 1 - 2.32T + 47T^{2} \)
53 \( 1 + 9.91iT - 53T^{2} \)
59 \( 1 - 0.328T + 59T^{2} \)
61 \( 1 - 1.53T + 61T^{2} \)
67 \( 1 - 9.98iT - 67T^{2} \)
71 \( 1 - 5.02T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 + 2.44iT - 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + 4.49iT - 89T^{2} \)
97 \( 1 - 5.62T + 97T^{2} \)
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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

−9.513752193690629716520337438159, −8.697840926117331926341879131975, −8.360365911743247155925128451521, −7.22647695225987744535995222963, −6.32878129214263296329235683170, −5.95591258430861443516800042971, −4.45204228387019119151782712140, −3.61289974739999931713618138164, −1.85371925341183831507747620820, −0.949192569465543654408655935160, 1.10496025024094419209468897044, 2.23074528319940904427462245633, 3.49871883805972371693767840756, 4.13544974089581345452391492013, 6.12967164080872496187471616531, 6.24341241093362262921493932334, 7.48840587365025838905246954837, 8.153617706971983895739387374964, 9.126085287691563284205642577674, 9.550936830707210160925012553869

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