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Properties

Label 2-42e2-63.5-c1-0-9
Degree $2$
Conductor $1764$
Sign $0.910 - 0.414i$
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  + (−1.70 − 0.287i)3-s + (0.266 + 0.462i)5-s + (2.83 + 0.983i)9-s + (3.39 + 1.96i)11-s + (−0.116 − 0.0674i)13-s + (−0.322 − 0.866i)15-s + (−2.16 − 3.74i)17-s + (1.93 + 1.11i)19-s + (1.70 − 0.983i)23-s + (2.35 − 4.08i)25-s + (−4.55 − 2.49i)27-s + (−5.16 + 2.98i)29-s + 0.924i·31-s + (−5.24 − 4.33i)33-s + (−3.89 + 6.75i)37-s + ⋯
L(s)  = 1  + (−0.986 − 0.166i)3-s + (0.119 + 0.206i)5-s + (0.944 + 0.327i)9-s + (1.02 + 0.591i)11-s + (−0.0324 − 0.0187i)13-s + (−0.0832 − 0.223i)15-s + (−0.524 − 0.908i)17-s + (0.442 + 0.255i)19-s + (0.355 − 0.205i)23-s + (0.471 − 0.816i)25-s + (−0.877 − 0.480i)27-s + (−0.959 + 0.553i)29-s + 0.165i·31-s + (−0.912 − 0.753i)33-s + (−0.641 + 1.11i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.910 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.287853794\)
\(L(\frac12)\) \(\approx\) \(1.287853794\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.287i)T \)
7 \( 1 \)
good5 \( 1 + (-0.266 - 0.462i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.39 - 1.96i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.116 + 0.0674i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.93 - 1.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.70 + 0.983i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.16 - 2.98i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.924iT - 31T^{2} \)
37 \( 1 + (3.89 - 6.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.59 - 7.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.24 - 5.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.08T + 47T^{2} \)
53 \( 1 + (-9.54 + 5.50i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.79T + 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 3.22iT - 71T^{2} \)
73 \( 1 + (0.329 - 0.190i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.20T + 79T^{2} \)
83 \( 1 + (1.28 + 2.21i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.56 - 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.6 + 7.89i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.552505697362925066032669080396, −8.621681505464803726898067021942, −7.50912710873115684408661554533, −6.79330399262922299947999743025, −6.34955615223044919673563925027, −5.19764254058002092037153792988, −4.62524870496779521839861129382, −3.53774725103523566860648696840, −2.15416088526027705291195562830, −0.966163684141777317742881926441, 0.74172284983049771124814873127, 1.93218964070625007688530356390, 3.62973730480393999216888070147, 4.19995836207708120506158211394, 5.44443373276322505529977051101, 5.80237672750030108875900453117, 6.86319446921922217354885406742, 7.36877710497375418128082326764, 8.791477783305786767249878402066, 9.107390026807851072243651446664

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