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Properties

Label 2-42e2-63.5-c1-0-38
Degree $2$
Conductor $1764$
Sign $-0.353 - 0.935i$
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  + (−0.106 − 1.72i)3-s + (−1.43 − 2.48i)5-s + (−2.97 + 0.369i)9-s + (−2.34 − 1.35i)11-s + (3.18 + 1.84i)13-s + (−4.14 + 2.74i)15-s + (−3.22 − 5.58i)17-s + (−2.73 − 1.58i)19-s + (2.59 − 1.49i)23-s + (−1.61 + 2.79i)25-s + (0.956 + 5.10i)27-s + (−2.48 + 1.43i)29-s + 9.54i·31-s + (−2.09 + 4.20i)33-s + (−1.70 + 2.95i)37-s + ⋯
L(s)  = 1  + (−0.0616 − 0.998i)3-s + (−0.641 − 1.11i)5-s + (−0.992 + 0.123i)9-s + (−0.708 − 0.408i)11-s + (0.884 + 0.510i)13-s + (−1.06 + 0.708i)15-s + (−0.781 − 1.35i)17-s + (−0.628 − 0.362i)19-s + (0.540 − 0.311i)23-s + (−0.322 + 0.558i)25-s + (0.184 + 0.982i)27-s + (−0.461 + 0.266i)29-s + 1.71i·31-s + (−0.364 + 0.732i)33-s + (−0.280 + 0.485i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.353 - 0.935i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3499725885\)
\(L(\frac12)\) \(\approx\) \(0.3499725885\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.106 + 1.72i)T \)
7 \( 1 \)
good5 \( 1 + (1.43 + 2.48i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.34 + 1.35i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.18 - 1.84i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.22 + 5.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.73 + 1.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 + 1.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.48 - 1.43i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.54iT - 31T^{2} \)
37 \( 1 + (1.70 - 2.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.794 + 1.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.67 + 8.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + (-2.16 + 1.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.67T + 59T^{2} \)
61 \( 1 - 0.654iT - 61T^{2} \)
67 \( 1 - 7.72T + 67T^{2} \)
71 \( 1 - 7.86iT - 71T^{2} \)
73 \( 1 + (11.0 - 6.39i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.19T + 79T^{2} \)
83 \( 1 + (-7.92 - 13.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.14 - 5.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.2 + 7.62i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.634048236035614530655782463129, −8.164727265851022814885699186421, −7.04485414868468522549606083662, −6.62117768709960252339454960854, −5.31800827724770456474091582325, −4.86304341058088964501136092849, −3.61260409616660023593731144302, −2.48399188318065451987563279776, −1.22615871118977040229541332031, −0.13882029228527745707486421090, 2.19377969410102643886334900880, 3.31882280218800866377998443780, 3.87344179598127886508323242368, 4.79076296824724711457626227597, 5.93862388430732256780234781149, 6.46091292446835676833528860408, 7.68449566096377185980139557755, 8.197437559314477984549497077144, 9.086543478743867314913722916552, 10.09920603548572888715364966710

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