Properties

Label 2-42e2-49.17-c0-0-0
Degree $2$
Conductor $1764$
Sign $0.725 + 0.687i$
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  + (−0.0747 − 0.997i)7-s + (−0.255 + 0.531i)13-s + (1.72 − 0.997i)19-s + (0.0747 − 0.997i)25-s + (−0.258 − 0.149i)31-s + (0.722 + 0.108i)37-s + (0.367 − 1.61i)43-s + (−0.988 + 0.149i)49-s + (−0.233 + 1.54i)61-s + (0.733 − 1.26i)67-s + (−1.85 − 0.139i)73-s + (0.988 + 1.71i)79-s + (0.548 + 0.215i)91-s + 0.867i·97-s + (0.766 + 0.825i)103-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)7-s + (−0.255 + 0.531i)13-s + (1.72 − 0.997i)19-s + (0.0747 − 0.997i)25-s + (−0.258 − 0.149i)31-s + (0.722 + 0.108i)37-s + (0.367 − 1.61i)43-s + (−0.988 + 0.149i)49-s + (−0.233 + 1.54i)61-s + (0.733 − 1.26i)67-s + (−1.85 − 0.139i)73-s + (0.988 + 1.71i)79-s + (0.548 + 0.215i)91-s + 0.867i·97-s + (0.766 + 0.825i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.725 + 0.687i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1585, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.725 + 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.124163738\)
\(L(\frac12)\) \(\approx\) \(1.124163738\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.0747 + 0.997i)T \)
good5 \( 1 + (-0.0747 + 0.997i)T^{2} \)
11 \( 1 + (0.365 - 0.930i)T^{2} \)
13 \( 1 + (0.255 - 0.531i)T + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.733 - 0.680i)T^{2} \)
19 \( 1 + (-1.72 + 0.997i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.258 + 0.149i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.722 - 0.108i)T + (0.955 + 0.294i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (-0.367 + 1.61i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.988 - 0.149i)T^{2} \)
53 \( 1 + (0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (0.233 - 1.54i)T + (-0.955 - 0.294i)T^{2} \)
67 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (1.85 + 0.139i)T + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.365 - 0.930i)T^{2} \)
97 \( 1 - 0.867iT - T^{2} \)
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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

−9.470185540800695671751527048425, −8.672252717498051454385735806704, −7.58870851159184348582226007345, −7.17387875151104328610973349395, −6.30332874288383732027884341154, −5.22438810459413306858984561750, −4.42433657415649138768150582191, −3.53667959650754562735619996800, −2.44579602251257062697658870291, −0.961629043350915996328353933977, 1.47885762481960495065198131957, 2.78369725215534963855468984894, 3.51055965222060543619863560258, 4.84973508206990868060106129527, 5.56945834401888681853302687484, 6.20276963657999062571833586294, 7.40897830667634109878474398594, 7.917440745498468835517162740377, 8.889727218421243268873475873211, 9.570454062444755690158039712509

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