Properties

Label 2-37026-1.1-c1-0-39
Degree $2$
Conductor $37026$
Sign $-1$
Analytic cond. $295.654$
Root an. cond. $17.1945$
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 + 4-s + 2·5-s + 8-s + 2·10-s + 2·13-s + 16-s + 17-s − 4·19-s + 2·20-s − 25-s + 2·26-s − 10·29-s + 8·31-s + 32-s + 34-s − 2·37-s − 4·38-s + 2·40-s + 10·41-s − 12·43-s − 7·49-s − 50-s + 2·52-s − 6·53-s − 10·58-s − 12·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 0.554·13-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.447·20-s − 1/5·25-s + 0.392·26-s − 1.85·29-s + 1.43·31-s + 0.176·32-s + 0.171·34-s − 0.328·37-s − 0.648·38-s + 0.316·40-s + 1.56·41-s − 1.82·43-s − 49-s − 0.141·50-s + 0.277·52-s − 0.824·53-s − 1.31·58-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37026\)    =    \(2 \cdot 3^{2} \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(295.654\)
Root analytic conductor: \(17.1945\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 37026,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.95301400910754, −14.68147434754901, −13.96303019121742, −13.53211524742031, −13.19951122936605, −12.64579543046357, −12.13497104132553, −11.37516824692944, −11.09725293918636, −10.36020287759205, −9.957599803632315, −9.314599391628568, −8.815003584345481, −7.985980518705446, −7.658667351356003, −6.656819144724891, −6.379919321231583, −5.798575803667200, −5.291241457428914, −4.577752653341313, −3.989046612377200, −3.289550490115519, −2.610620958088807, −1.846660678453263, −1.357617127457102, 0, 1.357617127457102, 1.846660678453263, 2.610620958088807, 3.289550490115519, 3.989046612377200, 4.577752653341313, 5.291241457428914, 5.798575803667200, 6.379919321231583, 6.656819144724891, 7.658667351356003, 7.985980518705446, 8.815003584345481, 9.314599391628568, 9.957599803632315, 10.36020287759205, 11.09725293918636, 11.37516824692944, 12.13497104132553, 12.64579543046357, 13.19951122936605, 13.53211524742031, 13.96303019121742, 14.68147434754901, 14.95301400910754

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