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Properties

Label 2-42e2-21.11-c2-0-18
Degree $2$
Conductor $1764$
Sign $0.177 + 0.984i$
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  + (−2.01 + 1.16i)5-s + (−7.70 − 4.44i)11-s − 8.58·13-s + (17.4 + 10.0i)17-s + (2.93 + 5.08i)19-s + (15.0 − 8.69i)23-s + (−9.79 + 16.9i)25-s + 19.0i·29-s + (2.35 − 4.07i)31-s + (−19.8 − 34.4i)37-s − 6.98i·41-s − 35.7·43-s + (63.5 − 36.6i)47-s + (−40.4 − 23.3i)53-s + 20.7·55-s + ⋯
L(s)  = 1  + (−0.403 + 0.232i)5-s + (−0.700 − 0.404i)11-s − 0.660·13-s + (1.02 + 0.591i)17-s + (0.154 + 0.267i)19-s + (0.654 − 0.377i)23-s + (−0.391 + 0.678i)25-s + 0.656i·29-s + (0.0759 − 0.131i)31-s + (−0.537 − 0.930i)37-s − 0.170i·41-s − 0.831·43-s + (1.35 − 0.780i)47-s + (−0.762 − 0.440i)53-s + 0.376·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.177 + 0.984i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ 0.177 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.091029276\)
\(L(\frac12)\) \(\approx\) \(1.091029276\)
\(L(2)\) 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 \( 1 + (2.01 - 1.16i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (7.70 + 4.44i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 8.58T + 169T^{2} \)
17 \( 1 + (-17.4 - 10.0i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.93 - 5.08i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-15.0 + 8.69i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 19.0iT - 841T^{2} \)
31 \( 1 + (-2.35 + 4.07i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (19.8 + 34.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 6.98iT - 1.68e3T^{2} \)
43 \( 1 + 35.7T + 1.84e3T^{2} \)
47 \( 1 + (-63.5 + 36.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (40.4 + 23.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (20.8 + 12.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (36.6 + 63.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-16.0 + 27.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 114. iT - 5.04e3T^{2} \)
73 \( 1 + (-20.6 + 35.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-13.3 - 23.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 136. iT - 6.88e3T^{2} \)
89 \( 1 + (-126. + 72.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 65.9T + 9.40e3T^{2} \)
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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

−8.925833211513051936079840711489, −7.979870130531945513402162478123, −7.52165271334889043484280937383, −6.61272538235114303010023130226, −5.57844748987131763861231627480, −4.97150725922806152764506827178, −3.72389076327441046443829348767, −3.06034666499668755255563224965, −1.81542590051898427621862449651, −0.33945423412825129150540254760, 0.974231474497562281604677898644, 2.41354126081058226096149712489, 3.28154603822496780679877376659, 4.48287549121496750127020889327, 5.09454013526699427537243008286, 6.00205870802019362269281673118, 7.15295505211592625929665921576, 7.64618660596359446383143229656, 8.402296863340439421585781489934, 9.373018673566269268159222204957

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