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Properties

Label 2-4410-105.104-c1-0-74
Degree $2$
Conductor $4410$
Sign $-0.750 + 0.660i$
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 + (−1.12 − 1.93i)5-s + 8-s + (−1.12 − 1.93i)10-s − 0.602i·11-s − 0.0571·13-s + 16-s − 0.347i·17-s − 4.92i·19-s + (−1.12 − 1.93i)20-s − 0.602i·22-s − 1.45·23-s + (−2.46 + 4.35i)25-s − 0.0571·26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.503 − 0.863i)5-s + 0.353·8-s + (−0.356 − 0.610i)10-s − 0.181i·11-s − 0.0158·13-s + 0.250·16-s − 0.0842i·17-s − 1.12i·19-s + (−0.251 − 0.431i)20-s − 0.128i·22-s − 0.304·23-s + (−0.492 + 0.870i)25-s − 0.0112·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.750 + 0.660i$
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: \((2,\ 4410,\ (\ :1/2),\ -0.750 + 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.686451294\)
\(L(\frac12)\) \(\approx\) \(1.686451294\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 + 1.93i)T \)
7 \( 1 \)
good11 \( 1 + 0.602iT - 11T^{2} \)
13 \( 1 + 0.0571T + 13T^{2} \)
17 \( 1 + 0.347iT - 17T^{2} \)
19 \( 1 + 4.92iT - 19T^{2} \)
23 \( 1 + 1.45T + 23T^{2} \)
29 \( 1 - 6.49iT - 29T^{2} \)
31 \( 1 + 5.05iT - 31T^{2} \)
37 \( 1 + 1.16iT - 37T^{2} \)
41 \( 1 + 2.64T + 41T^{2} \)
43 \( 1 + 12.0iT - 43T^{2} \)
47 \( 1 + 7.74iT - 47T^{2} \)
53 \( 1 + 5.55T + 53T^{2} \)
59 \( 1 - 0.890T + 59T^{2} \)
61 \( 1 + 5.72iT - 61T^{2} \)
67 \( 1 + 0.110iT - 67T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + 0.914T + 73T^{2} \)
79 \( 1 - 1.38T + 79T^{2} \)
83 \( 1 - 0.429iT - 83T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−8.095999498540326157264996607595, −7.20787855863068120337807636630, −6.69903160298277782395694800014, −5.54311729222568159057599077152, −5.17811240964972539070489165950, −4.28413536591275736834435674028, −3.67545772081737850377103295220, −2.68738994009608626215729241051, −1.60163854287679424204037432705, −0.34842747302597086615174483488, 1.49471172266252491938892420565, 2.59521827930245215974604040285, 3.31483659056986897356620252989, 4.09872934365874598796606092704, 4.76089968829428687420097430638, 5.87853402970063736780373511617, 6.31877336875013909521600812825, 7.11526370675565894990026830547, 7.85687416494223911397868337131, 8.288256115103509834953935698739

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