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Properties

Label 2-42e2-84.59-c0-0-2
Degree $2$
Conductor $1764$
Sign $-0.413 - 0.910i$
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 + (0.923 + 1.60i)5-s + 0.999i·8-s + (−1.60 − 0.923i)10-s + 0.765i·13-s + (−0.5 − 0.866i)16-s + (−0.382 + 0.662i)17-s + 1.84·20-s + (−1.20 + 2.09i)25-s + (−0.382 − 0.662i)26-s + (0.866 + 0.499i)32-s − 0.765i·34-s + (−0.707 − 1.22i)37-s + (−1.60 + 0.923i)40-s + 0.765·41-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.923 + 1.60i)5-s + 0.999i·8-s + (−1.60 − 0.923i)10-s + 0.765i·13-s + (−0.5 − 0.866i)16-s + (−0.382 + 0.662i)17-s + 1.84·20-s + (−1.20 + 2.09i)25-s + (−0.382 − 0.662i)26-s + (0.866 + 0.499i)32-s − 0.765i·34-s + (−0.707 − 1.22i)37-s + (−1.60 + 0.923i)40-s + 0.765·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.413 - 0.910i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ -0.413 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8429109180\)
\(L(\frac12)\) \(\approx\) \(0.8429109180\)
\(L(1)\) 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 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - 0.765iT - T^{2} \)
17 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 0.765T + T^{2} \)
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 \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.84iT - T^{2} \)
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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

−9.690907288258332643245851916780, −9.112675925622376113226001953849, −8.114419957337817731373389286175, −7.20667561940764863530675549710, −6.64247607413064921161971749778, −6.09879249611845345304274868716, −5.22258793155786761799572264624, −3.74500194172283528032333783069, −2.49490116475963109977274232561, −1.81146344584377180028351987538, 0.857297185712590379902205919648, 1.88203065044602014363585955014, 2.94960379225934915968711266153, 4.28379365450596330561295762791, 5.13368044988761498334716832833, 5.99920618460051704089451217510, 7.00635064601897715304066386048, 8.093967962894371485037529882259, 8.529207501125716095487817596266, 9.368239203677408910131623686353

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