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Properties

Label 2-42e2-12.11-c1-0-10
Degree $2$
Conductor $1764$
Sign $-0.971 - 0.236i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (1.16 + 0.795i)2-s + (0.735 + 1.85i)4-s − 0.665i·5-s + (−0.619 + 2.75i)8-s + (0.529 − 0.778i)10-s − 2.07·11-s − 5.55·13-s + (−2.91 + 2.73i)16-s + 2.16i·17-s + 4.49i·19-s + (1.23 − 0.489i)20-s + (−2.43 − 1.65i)22-s − 4.28·23-s + 4.55·25-s + (−6.49 − 4.41i)26-s + ⋯
L(s)  = 1  + (0.826 + 0.562i)2-s + (0.367 + 0.929i)4-s − 0.297i·5-s + (−0.218 + 0.975i)8-s + (0.167 − 0.246i)10-s − 0.627·11-s − 1.54·13-s + (−0.729 + 0.683i)16-s + 0.524i·17-s + 1.03i·19-s + (0.276 − 0.109i)20-s + (−0.518 − 0.352i)22-s − 0.892·23-s + 0.911·25-s + (−1.27 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.971 - 0.236i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.971 - 0.236i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.511635685\)
\(L(\frac12)\) \(\approx\) \(1.511635685\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 0.795i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.665iT - 5T^{2} \)
11 \( 1 + 2.07T + 11T^{2} \)
13 \( 1 + 5.55T + 13T^{2} \)
17 \( 1 - 2.16iT - 17T^{2} \)
19 \( 1 - 4.49iT - 19T^{2} \)
23 \( 1 + 4.28T + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 - 6.61iT - 31T^{2} \)
37 \( 1 + 5.43T + 37T^{2} \)
41 \( 1 - 5.69iT - 41T^{2} \)
43 \( 1 - 2.11iT - 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 10.6iT - 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 - 0.615T + 61T^{2} \)
67 \( 1 - 14.0iT - 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 0.824iT - 79T^{2} \)
83 \( 1 + 6.36T + 83T^{2} \)
89 \( 1 + 12.6iT - 89T^{2} \)
97 \( 1 + 0.824T + 97T^{2} \)
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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

−9.742076038394884722672130677634, −8.445131315265911117196057069314, −8.113757779668059031444198093558, −7.13387719482952538156067543091, −6.50698489624667706936671205530, −5.34678167351412231656685881977, −5.00813417262083376418196444100, −3.94974858393761581422339126670, −2.98230651167896567919851727794, −1.91950071872743811978946926981, 0.38228535907633977252529605905, 2.20141982132110899645995472719, 2.73515321378192857812259653857, 3.86997116346718119551069574715, 4.91562538955221853417812210156, 5.32381057139616043530151863580, 6.52925233655926799717547958216, 7.13839534558718087073517836424, 8.001185461119240592412459366463, 9.291895207299925061339514585283

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