Properties

Label 2-42e2-84.47-c0-0-5
Degree $2$
Conductor $1764$
Sign $-0.329 - 0.944i$
Analytic cond. $0.880350$
Root an. cond. $0.938270$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.382 + 0.662i)5-s + 0.999i·8-s + (−0.662 + 0.382i)10-s + 1.84i·13-s + (−0.5 + 0.866i)16-s + (−0.923 − 1.60i)17-s − 0.765·20-s + (0.207 + 0.358i)25-s + (−0.923 + 1.60i)26-s + (−0.866 + 0.499i)32-s − 1.84i·34-s + (0.707 − 1.22i)37-s + (−0.662 − 0.382i)40-s + 1.84·41-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.382 + 0.662i)5-s + 0.999i·8-s + (−0.662 + 0.382i)10-s + 1.84i·13-s + (−0.5 + 0.866i)16-s + (−0.923 − 1.60i)17-s − 0.765·20-s + (0.207 + 0.358i)25-s + (−0.923 + 1.60i)26-s + (−0.866 + 0.499i)32-s − 1.84i·34-s + (0.707 − 1.22i)37-s + (−0.662 − 0.382i)40-s + 1.84·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.329 - 0.944i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ -0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.667505686\)
\(L(\frac12)\) \(\approx\) \(1.667505686\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.84iT - T^{2} \)
17 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.84T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 0.765iT - T^{2} \)
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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

−9.406036253250504358086962126000, −9.026218481918852137830391613825, −7.80333123671907175471272098558, −7.10208679579773178768928915530, −6.70209376955859749358618792907, −5.75333818533394771824086213048, −4.62333128495802153515283665367, −4.14195188894714314177308490023, −2.99947708517848359050068665488, −2.14780905943798996472039927282, 0.986698763097903668707774752801, 2.36321983537022563119375797175, 3.39964242549572195257889258875, 4.26444073921001985255898916935, 5.01645283890148552296057764828, 5.89475190291607385504853077715, 6.54625041447735195470669254310, 7.83144316563523199415323965705, 8.331901937635687884491201020522, 9.376265564540767641262182437768

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