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Properties

Label 2-39326-1.1-c1-0-13
Degree $2$
Conductor $39326$
Sign $-1$
Analytic cond. $314.019$
Root an. cond. $17.7206$
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 + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s + 2·12-s − 4·13-s + 14-s + 16-s + 6·17-s + 18-s − 2·19-s + 2·21-s + 2·24-s − 5·25-s − 4·26-s − 4·27-s + 28-s − 6·29-s + 4·31-s + 32-s + 6·34-s + 36-s + 2·37-s − 2·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.436·21-s + 0.408·24-s − 25-s − 0.784·26-s − 0.769·27-s + 0.188·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39326 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39326 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39326\)    =    \(2 \cdot 7 \cdot 53^{2}\)
Sign: $-1$
Analytic conductor: \(314.019\)
Root analytic conductor: \(17.7206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 39326,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
53 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.92577722722066, −14.55591847458322, −14.01107249625630, −13.75692018859631, −13.03470375027526, −12.57924081549194, −12.03081430828056, −11.53800277295444, −10.99900759089642, −10.14844507052972, −9.796672651142235, −9.308972033098977, −8.538339756907417, −7.988951928942078, −7.608946560520499, −7.189650630268929, −6.243342528426702, −5.735892317051575, −5.085001008922231, −4.494160154590321, −3.786040638551733, −3.263344999470288, −2.657000641368070, −2.035925368838017, −1.386840074537113, 0, 1.386840074537113, 2.035925368838017, 2.657000641368070, 3.263344999470288, 3.786040638551733, 4.494160154590321, 5.085001008922231, 5.735892317051575, 6.243342528426702, 7.189650630268929, 7.608946560520499, 7.988951928942078, 8.538339756907417, 9.308972033098977, 9.796672651142235, 10.14844507052972, 10.99900759089642, 11.53800277295444, 12.03081430828056, 12.57924081549194, 13.03470375027526, 13.75692018859631, 14.01107249625630, 14.55591847458322, 14.92577722722066

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