Properties

Label 2-61710-1.1-c1-0-49
Degree $2$
Conductor $61710$
Sign $-1$
Analytic cond. $492.756$
Root an. cond. $22.1981$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s − 12-s + 2·14-s + 15-s + 16-s − 17-s + 18-s − 20-s − 2·21-s − 4·23-s − 24-s + 25-s − 27-s + 2·28-s − 2·29-s + 30-s + 4·31-s + 32-s − 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.223·20-s − 0.436·21-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.182·30-s + 0.718·31-s + 0.176·32-s − 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61710\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(492.756\)
Root analytic conductor: \(22.1981\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{61710} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 61710,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 \)
17 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.63031867177991, −13.99510451523658, −13.60267591815603, −12.84178480172067, −12.59870929804820, −11.91773049864323, −11.56442050297616, −11.13010922362602, −10.74155420942737, −9.937568103627447, −9.703975180773519, −8.530650339947738, −8.448220269938256, −7.616662022036926, −7.213329130099336, −6.623109373269739, −5.988351421343137, −5.501975667331152, −4.940527565110534, −4.315423558340183, −4.013402235801823, −3.199514709761634, −2.442283590078994, −1.754309005162276, −1.010798257633693, 0, 1.010798257633693, 1.754309005162276, 2.442283590078994, 3.199514709761634, 4.013402235801823, 4.315423558340183, 4.940527565110534, 5.501975667331152, 5.988351421343137, 6.623109373269739, 7.213329130099336, 7.616662022036926, 8.448220269938256, 8.530650339947738, 9.703975180773519, 9.937568103627447, 10.74155420942737, 11.13010922362602, 11.56442050297616, 11.91773049864323, 12.59870929804820, 12.84178480172067, 13.60267591815603, 13.99510451523658, 14.63031867177991

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