Properties

Label 2-3330-5.4-c1-0-63
Degree $2$
Conductor $3330$
Sign $-0.316 + 0.948i$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
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  i·2-s − 4-s + (0.707 − 2.12i)5-s − 0.414i·7-s + i·8-s + (−2.12 − 0.707i)10-s + 3.58·11-s − 3.24i·13-s − 0.414·14-s + 16-s + 6.41i·17-s + 7.82·19-s + (−0.707 + 2.12i)20-s − 3.58i·22-s − 3i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.316 − 0.948i)5-s − 0.156i·7-s + 0.353i·8-s + (−0.670 − 0.223i)10-s + 1.08·11-s − 0.899i·13-s − 0.110·14-s + 0.250·16-s + 1.55i·17-s + 1.79·19-s + (−0.158 + 0.474i)20-s − 0.764i·22-s − 0.625i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $-0.316 + 0.948i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ -0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.126306092\)
\(L(\frac12)\) \(\approx\) \(2.126306092\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (-0.707 + 2.12i)T \)
37 \( 1 - iT \)
good7 \( 1 + 0.414iT - 7T^{2} \)
11 \( 1 - 3.58T + 11T^{2} \)
13 \( 1 + 3.24iT - 13T^{2} \)
17 \( 1 - 6.41iT - 17T^{2} \)
19 \( 1 - 7.82T + 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 - 7.65T + 31T^{2} \)
41 \( 1 - 9.41T + 41T^{2} \)
43 \( 1 + 6.48iT - 43T^{2} \)
47 \( 1 - 7.89iT - 47T^{2} \)
53 \( 1 + 1.82iT - 53T^{2} \)
59 \( 1 - 1.07T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + 4.82iT - 67T^{2} \)
71 \( 1 - 6.58T + 71T^{2} \)
73 \( 1 - 7.82iT - 73T^{2} \)
79 \( 1 + 3.75T + 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + 18.7iT - 97T^{2} \)
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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

−8.422273481806888590390445428081, −7.996214818308718025897825499016, −6.91884896837383320930239674332, −5.81929457276584053449617401901, −5.43894199230826200695674049974, −4.28432362210170583100084624115, −3.80906244822043070276087137758, −2.69457423616743438358077326095, −1.50116643417138729373587037472, −0.821658732579419575825030582974, 1.11730328185203863398007475186, 2.45364399517417283765646056628, 3.39963307805160160114423976814, 4.22543932792871929260577073227, 5.27848962296933944212535159378, 5.89415598897672982755498625803, 6.81695663087047428760103504086, 7.16319785722791459015456354103, 7.84069419316338519530388621522, 9.044085013236539428392505381180

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