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Properties

Label 2-42e2-12.11-c1-0-23
Degree $2$
Conductor $1764$
Sign $-0.345 - 0.938i$
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  + (0.658 + 1.25i)2-s + (−1.13 + 1.64i)4-s − 2.08i·5-s + (−2.80 − 0.333i)8-s + (2.60 − 1.37i)10-s − 4.26·11-s + 4.80·13-s + (−1.43 − 3.73i)16-s + 3.20i·17-s + 2.81i·19-s + (3.42 + 2.35i)20-s + (−2.80 − 5.34i)22-s + 4.66·23-s + 0.669·25-s + (3.16 + 6.01i)26-s + ⋯
L(s)  = 1  + (0.465 + 0.885i)2-s + (−0.566 + 0.823i)4-s − 0.930i·5-s + (−0.993 − 0.117i)8-s + (0.823 − 0.433i)10-s − 1.28·11-s + 1.33·13-s + (−0.357 − 0.933i)16-s + 0.777i·17-s + 0.646i·19-s + (0.766 + 0.527i)20-s + (−0.599 − 1.13i)22-s + 0.971·23-s + 0.133·25-s + (0.620 + 1.17i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.345 - 0.938i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.345 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.765366997\)
\(L(\frac12)\) \(\approx\) \(1.765366997\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.658 - 1.25i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.08iT - 5T^{2} \)
11 \( 1 + 4.26T + 11T^{2} \)
13 \( 1 - 4.80T + 13T^{2} \)
17 \( 1 - 3.20iT - 17T^{2} \)
19 \( 1 - 2.81iT - 19T^{2} \)
23 \( 1 - 4.66T + 23T^{2} \)
29 \( 1 - 3.87iT - 29T^{2} \)
31 \( 1 - 10.2iT - 31T^{2} \)
37 \( 1 - 0.273T + 37T^{2} \)
41 \( 1 + 0.387iT - 41T^{2} \)
43 \( 1 - 0.907iT - 43T^{2} \)
47 \( 1 - 7.85T + 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 + 3.70T + 59T^{2} \)
61 \( 1 - 8.02T + 61T^{2} \)
67 \( 1 - 1.40iT - 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 0.826iT - 79T^{2} \)
83 \( 1 + 5.69T + 83T^{2} \)
89 \( 1 + 3.02iT - 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−9.026715550866669884554486584228, −8.676511218855393544431455690589, −8.013956433075092287681435794809, −7.16515779949349795389304098754, −6.19981479692909604693593133417, −5.43963791146277535316196987396, −4.85564264531243154861744925046, −3.86486259079596079245271794700, −2.94692650874088802418691910454, −1.21478243964803465166301796354, 0.64136548603388047988013169383, 2.29345170181680519184441103190, 2.92361374052802168333415053922, 3.82460200151783102453839511948, 4.87194656133959237345509599511, 5.69478701475795042531112191989, 6.50078307221484836619969883438, 7.42789710467054303189433234429, 8.401609154466295365510380439236, 9.264312419912561123391433683950

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