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Properties

Label 2-1650-165.164-c1-0-44
Degree $2$
Conductor $1650$
Sign $0.260 + 0.965i$
Analytic cond. $13.1753$
Root an. cond. $3.62978$
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 + (−0.117 + 1.72i)3-s − 4-s + (1.72 + 0.117i)6-s + 0.723·7-s + i·8-s + (−2.97 − 0.404i)9-s + (−2.32 + 2.36i)11-s + (0.117 − 1.72i)12-s − 4.17·13-s − 0.723i·14-s + 16-s − 4.49i·17-s + (−0.404 + 2.97i)18-s − 7.94i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.0675 + 0.997i)3-s − 0.5·4-s + (0.705 + 0.0477i)6-s + 0.273·7-s + 0.353i·8-s + (−0.990 − 0.134i)9-s + (−0.700 + 0.713i)11-s + (0.0337 − 0.498i)12-s − 1.15·13-s − 0.193i·14-s + 0.250·16-s − 1.09i·17-s + (−0.0953 + 0.700i)18-s − 1.82i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(13.1753\)
Root analytic conductor: \(3.62978\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1650} (1649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1650,\ (\ :1/2),\ 0.260 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.098126541\)
\(L(\frac12)\) \(\approx\) \(1.098126541\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.117 - 1.72i)T \)
5 \( 1 \)
11 \( 1 + (2.32 - 2.36i)T \)
good7 \( 1 - 0.723T + 7T^{2} \)
13 \( 1 + 4.17T + 13T^{2} \)
17 \( 1 + 4.49iT - 17T^{2} \)
19 \( 1 + 7.94iT - 19T^{2} \)
23 \( 1 - 8.91T + 23T^{2} \)
29 \( 1 - 4.98T + 29T^{2} \)
31 \( 1 + 5.85T + 31T^{2} \)
37 \( 1 + 5.54iT - 37T^{2} \)
41 \( 1 - 3.53T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 - 4.44T + 47T^{2} \)
53 \( 1 - 2.41T + 53T^{2} \)
59 \( 1 - 9.71iT - 59T^{2} \)
61 \( 1 - 2.59iT - 61T^{2} \)
67 \( 1 + 9.24iT - 67T^{2} \)
71 \( 1 + 3.54iT - 71T^{2} \)
73 \( 1 - 4.56T + 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 + 2.10iT - 89T^{2} \)
97 \( 1 - 8.27iT - 97T^{2} \)
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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

−9.209858376957065385616934076900, −8.976824877754442342753547487691, −7.59245887396753782739699216951, −7.00793572120850637081656744902, −5.42629158745843232139065574210, −4.88839838743372078744666148917, −4.37009315272983167449666064514, −2.89320129885354683212998858087, −2.51714661709147801143245260106, −0.48345378165206270929817310047, 1.12426172286315738350836363390, 2.45473898273040668014930829026, 3.56853203797949643990171380632, 4.92978613152578962618001317312, 5.62492876843059739342262819946, 6.34421396180474545561544118872, 7.23862382704027705557898584708, 7.899594123473216985530427580933, 8.389497902657683676707999649822, 9.274820427211129488741467425928

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