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Properties

Label 2-1710-95.37-c1-0-15
Degree $2$
Conductor $1710$
Sign $0.993 - 0.111i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−1.89 + 1.18i)5-s + (0.705 + 0.705i)7-s + (0.707 + 0.707i)8-s + (0.498 − 2.17i)10-s − 1.32·11-s + (−0.741 − 0.741i)13-s − 0.997·14-s − 1.00·16-s + (−2.17 − 2.17i)17-s + (0.939 − 4.25i)19-s + (1.18 + 1.89i)20-s + (0.939 − 0.939i)22-s + (1.08 − 1.08i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.846 + 0.531i)5-s + (0.266 + 0.266i)7-s + (0.250 + 0.250i)8-s + (0.157 − 0.689i)10-s − 0.400·11-s + (−0.205 − 0.205i)13-s − 0.266·14-s − 0.250·16-s + (−0.527 − 0.527i)17-s + (0.215 − 0.976i)19-s + (0.265 + 0.423i)20-s + (0.200 − 0.200i)22-s + (0.225 − 0.225i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.993 - 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9289983629\)
\(L(\frac12)\) \(\approx\) \(0.9289983629\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (1.89 - 1.18i)T \)
19 \( 1 + (-0.939 + 4.25i)T \)
good7 \( 1 + (-0.705 - 0.705i)T + 7iT^{2} \)
11 \( 1 + 1.32T + 11T^{2} \)
13 \( 1 + (0.741 + 0.741i)T + 13iT^{2} \)
17 \( 1 + (2.17 + 2.17i)T + 17iT^{2} \)
23 \( 1 + (-1.08 + 1.08i)T - 23iT^{2} \)
29 \( 1 - 5.95T + 29T^{2} \)
31 \( 1 - 5.95iT - 31T^{2} \)
37 \( 1 + (1.33 - 1.33i)T - 37iT^{2} \)
41 \( 1 - 0.531iT - 41T^{2} \)
43 \( 1 + (-1.53 + 1.53i)T - 43iT^{2} \)
47 \( 1 + (4.13 + 4.13i)T + 47iT^{2} \)
53 \( 1 + (-5.48 - 5.48i)T + 53iT^{2} \)
59 \( 1 - 3.53T + 59T^{2} \)
61 \( 1 - 3.41T + 61T^{2} \)
67 \( 1 + (1.99 - 1.99i)T - 67iT^{2} \)
71 \( 1 + 8.71iT - 71T^{2} \)
73 \( 1 + (5.83 - 5.83i)T - 73iT^{2} \)
79 \( 1 - 9.31T + 79T^{2} \)
83 \( 1 + (-9.50 + 9.50i)T - 83iT^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (-1.11 + 1.11i)T - 97iT^{2} \)
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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

−9.071936555028714416006390060763, −8.546191874216561789660043372969, −7.73309661999335786145720567859, −7.03313606226755497845504637323, −6.44972005345423841431896117798, −5.18157834458560141633104515876, −4.61538469197169753933657912451, −3.27543169793386069089361142916, −2.34465984433537219578781761998, −0.58360206796172013868757043488, 0.869123571132206243527211192525, 2.11251841491107657822581339742, 3.38800656323186411366555939328, 4.20875307176276853061250913825, 4.97903852703752398203285099902, 6.16128571897205366671387497469, 7.26546223269409868359412251606, 7.916563786422326969195633952612, 8.470343083696243784342939122503, 9.292112133628018188281947288162

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