Properties

Label 2-4410-105.104-c1-0-29
Degree $2$
Conductor $4410$
Sign $0.126 - 0.991i$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
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-s + 4-s + (2.12 + 0.705i)5-s + 8-s + (2.12 + 0.705i)10-s + 3.38i·11-s − 0.920·13-s + 16-s + 7.01i·17-s + 0.709i·19-s + (2.12 + 0.705i)20-s + 3.38i·22-s − 2.11·23-s + (4.00 + 2.99i)25-s − 0.920·26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.948 + 0.315i)5-s + 0.353·8-s + (0.671 + 0.223i)10-s + 1.01i·11-s − 0.255·13-s + 0.250·16-s + 1.70i·17-s + 0.162i·19-s + (0.474 + 0.157i)20-s + 0.720i·22-s − 0.441·23-s + (0.800 + 0.598i)25-s − 0.180·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.126 - 0.991i$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4410} (4409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 0.126 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.335561267\)
\(L(\frac12)\) \(\approx\) \(3.335561267\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + (-2.12 - 0.705i)T \)
7 \( 1 \)
good11 \( 1 - 3.38iT - 11T^{2} \)
13 \( 1 + 0.920T + 13T^{2} \)
17 \( 1 - 7.01iT - 17T^{2} \)
19 \( 1 - 0.709iT - 19T^{2} \)
23 \( 1 + 2.11T + 23T^{2} \)
29 \( 1 - 0.0694iT - 29T^{2} \)
31 \( 1 + 4.09iT - 31T^{2} \)
37 \( 1 + 2.81iT - 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 7.35iT - 43T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 - 2.79T + 53T^{2} \)
59 \( 1 - 5.80T + 59T^{2} \)
61 \( 1 - 5.18iT - 61T^{2} \)
67 \( 1 + 6.80iT - 67T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + 6.18T + 73T^{2} \)
79 \( 1 - 4.57T + 79T^{2} \)
83 \( 1 - 8.39iT - 83T^{2} \)
89 \( 1 + 0.636T + 89T^{2} \)
97 \( 1 + 3.00T + 97T^{2} \)
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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

−8.412308376511543465993880244429, −7.67877770742496166099198669834, −6.86805785222493899994049783428, −6.21176027393100757455601434505, −5.68535595336673051673953600839, −4.78094189392907051907274779902, −4.08356396115471905388177189721, −3.13478163206933284818084455679, −2.13762517709946004716555142836, −1.57225343040675776267535521939, 0.66499077919999818777458415335, 1.90296647086378331354444032427, 2.77755278196965944517289182994, 3.51403057599684308548184439443, 4.64018573997670079777227161485, 5.32177884320677875343219333043, 5.72031954091012837453114875081, 6.78327930062555110412503959697, 7.09203973051717333232103737707, 8.368316101278729236558753646813

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