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Properties

Label 2-42e2-7.6-c2-0-11
Degree $2$
Conductor $1764$
Sign $-0.156 - 0.987i$
Analytic cond. $48.0655$
Root an. cond. $6.93293$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.15i·5-s − 1.95·11-s − 0.317i·13-s − 18.4i·17-s + 6.94i·19-s + 36.8·23-s − 12.8·25-s + 40.7·29-s + 30.2i·31-s − 13.6·37-s + 30.1i·41-s + 41.7·43-s + 48.6i·47-s + 3.90·53-s − 12.0i·55-s + ⋯
L(s)  = 1  + 1.23i·5-s − 0.177·11-s − 0.0243i·13-s − 1.08i·17-s + 0.365i·19-s + 1.60·23-s − 0.515·25-s + 1.40·29-s + 0.976i·31-s − 0.370·37-s + 0.735i·41-s + 0.972·43-s + 1.03i·47-s + 0.0736·53-s − 0.218i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.156 - 0.987i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.156 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.787500285\)
\(L(\frac12)\) \(\approx\) \(1.787500285\)
\(L(2)\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 - 6.15iT - 25T^{2} \)
11 \( 1 + 1.95T + 121T^{2} \)
13 \( 1 + 0.317iT - 169T^{2} \)
17 \( 1 + 18.4iT - 289T^{2} \)
19 \( 1 - 6.94iT - 361T^{2} \)
23 \( 1 - 36.8T + 529T^{2} \)
29 \( 1 - 40.7T + 841T^{2} \)
31 \( 1 - 30.2iT - 961T^{2} \)
37 \( 1 + 13.6T + 1.36e3T^{2} \)
41 \( 1 - 30.1iT - 1.68e3T^{2} \)
43 \( 1 - 41.7T + 1.84e3T^{2} \)
47 \( 1 - 48.6iT - 2.20e3T^{2} \)
53 \( 1 - 3.90T + 2.80e3T^{2} \)
59 \( 1 + 24.0iT - 3.48e3T^{2} \)
61 \( 1 - 79.8iT - 3.72e3T^{2} \)
67 \( 1 + 91.3T + 4.48e3T^{2} \)
71 \( 1 + 75.7T + 5.04e3T^{2} \)
73 \( 1 - 132. iT - 5.32e3T^{2} \)
79 \( 1 + 25.1T + 6.24e3T^{2} \)
83 \( 1 + 84.9iT - 6.88e3T^{2} \)
89 \( 1 + 128. iT - 7.92e3T^{2} \)
97 \( 1 - 4.05iT - 9.40e3T^{2} \)
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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

−9.339378539466196112354563054759, −8.590109013247577673768952510531, −7.52001401701991141193745272555, −7.00303303201745108095886740037, −6.29847088303292481382896423358, −5.26803162116622719388170149311, −4.39231146148174373734128238671, −3.02799017371576129185794418077, −2.77198388360595304288297162286, −1.14219833787540929477929886745, 0.53456093072238995077945623037, 1.55569590505110534112640910152, 2.82714083334498221187949179651, 4.03697835960250232243124508683, 4.80396132996523346003092457029, 5.51470368176713022497953549831, 6.45522382804303995291785683873, 7.39027029252390298230631879725, 8.323946089408795093536512478206, 8.816995903327718449217492885648

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