Properties

Label 2-21e2-441.5-c1-0-12
Degree $2$
Conductor $441$
Sign $-0.210 - 0.977i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
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  + (−2.16 + 1.72i)2-s + (0.581 − 1.63i)3-s + (1.25 − 5.51i)4-s + (2.91 + 1.98i)5-s + (1.55 + 4.53i)6-s + (−2.07 + 1.64i)7-s + (4.39 + 9.11i)8-s + (−2.32 − 1.89i)9-s + (−9.72 + 0.728i)10-s + (0.128 + 0.851i)11-s + (−8.26 − 5.25i)12-s + (0.596 + 3.95i)13-s + (1.64 − 7.13i)14-s + (4.93 − 3.59i)15-s + (−15.0 − 7.24i)16-s + (−1.51 + 1.40i)17-s + ⋯
L(s)  = 1  + (−1.52 + 1.22i)2-s + (0.335 − 0.942i)3-s + (0.629 − 2.75i)4-s + (1.30 + 0.887i)5-s + (0.636 + 1.85i)6-s + (−0.783 + 0.621i)7-s + (1.55 + 3.22i)8-s + (−0.774 − 0.632i)9-s + (−3.07 + 0.230i)10-s + (0.0387 + 0.256i)11-s + (−2.38 − 1.51i)12-s + (0.165 + 1.09i)13-s + (0.439 − 1.90i)14-s + (1.27 − 0.929i)15-s + (−3.76 − 1.81i)16-s + (−0.368 + 0.341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.210 - 0.977i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.210 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487875 + 0.604298i\)
\(L(\frac12)\) \(\approx\) \(0.487875 + 0.604298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.581 + 1.63i)T \)
7 \( 1 + (2.07 - 1.64i)T \)
good2 \( 1 + (2.16 - 1.72i)T + (0.445 - 1.94i)T^{2} \)
5 \( 1 + (-2.91 - 1.98i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.128 - 0.851i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-0.596 - 3.95i)T + (-12.4 + 3.83i)T^{2} \)
17 \( 1 + (1.51 - 1.40i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-1.72 - 0.995i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.379 - 1.23i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-6.03 - 6.50i)T + (-2.16 + 28.9i)T^{2} \)
31 \( 1 + 2.77iT - 31T^{2} \)
37 \( 1 + (-1.25 - 0.387i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-0.381 + 5.08i)T + (-40.5 - 6.11i)T^{2} \)
43 \( 1 + (0.323 + 4.31i)T + (-42.5 + 6.40i)T^{2} \)
47 \( 1 + (-2.93 - 3.67i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-3.02 - 9.81i)T + (-43.7 + 29.8i)T^{2} \)
59 \( 1 + (3.49 + 1.68i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (3.17 - 0.723i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + 6.59T + 67T^{2} \)
71 \( 1 + (0.0685 + 0.0156i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (0.591 - 3.92i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 - 8.89T + 79T^{2} \)
83 \( 1 + (-11.9 - 1.80i)T + (79.3 + 24.4i)T^{2} \)
89 \( 1 + (0.985 + 2.50i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-0.845 + 0.488i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.91960310123435047057077971386, −10.07557596694517480901895295595, −9.188881744596341879488842490045, −8.883853890477664302964434908712, −7.54663604842894958473435101020, −6.70911346475232406001491468724, −6.33356394886578343441295645418, −5.53777930912457835370295675794, −2.56573700722067912392559334943, −1.60964524659514122371890282182, 0.794999608805949313799312208780, 2.45696845518746486333831320145, 3.37890503124496108609003615174, 4.71347310185846710448101632200, 6.27637373448129478164722603651, 7.78730374700877228102972962064, 8.674835045940233457200071901855, 9.309144969795408144942133895610, 10.03511822113768438757650906314, 10.32999412698848875601928753711

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