Properties

Degree 2
Conductor 2
Sign $-1$
Motivic weight 33
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.55e4·2-s − 1.33e8·3-s + 4.29e9·4-s + 5.38e11·5-s + 8.71e12·6-s − 3.33e13·7-s − 2.81e14·8-s + 1.21e16·9-s − 3.53e16·10-s − 8.58e16·11-s − 5.71e17·12-s + 1.14e18·13-s + 2.18e18·14-s − 7.16e19·15-s + 1.84e19·16-s − 1.39e20·17-s − 7.95e20·18-s + 8.06e19·19-s + 2.31e21·20-s + 4.43e21·21-s + 5.62e21·22-s − 1.41e22·23-s + 3.74e22·24-s + 1.73e23·25-s − 7.49e22·26-s − 8.74e23·27-s − 1.43e23·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.78·3-s + 1/2·4-s + 1.57·5-s + 1.26·6-s − 0.379·7-s − 0.353·8-s + 2.18·9-s − 1.11·10-s − 0.563·11-s − 0.891·12-s + 0.476·13-s + 0.268·14-s − 2.81·15-s + 1/4·16-s − 0.693·17-s − 1.54·18-s + 0.0641·19-s + 0.789·20-s + 0.676·21-s + 0.398·22-s − 0.480·23-s + 0.630·24-s + 1.49·25-s − 0.337·26-s − 2.10·27-s − 0.189·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(33\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 2,\ (\ :33/2),\ -1)\)
\(L(17)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{35}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + p^{16} T \)
good3 \( 1 + 547348 p^{5} T + p^{33} T^{2} \)
5 \( 1 - 21551965302 p^{2} T + p^{33} T^{2} \)
7 \( 1 + 4763901578824 p T + p^{33} T^{2} \)
11 \( 1 + 7807478511398748 p T + p^{33} T^{2} \)
13 \( 1 - 88004154528915782 p T + p^{33} T^{2} \)
17 \( 1 + 8183157392316801294 p T + p^{33} T^{2} \)
19 \( 1 - 4247105272322643740 p T + p^{33} T^{2} \)
23 \( 1 + \)\(61\!\cdots\!28\)\( p T + p^{33} T^{2} \)
29 \( 1 + \)\(56\!\cdots\!10\)\( p T + p^{33} T^{2} \)
31 \( 1 + \)\(61\!\cdots\!88\)\( p T + p^{33} T^{2} \)
37 \( 1 + \)\(96\!\cdots\!98\)\( T + p^{33} T^{2} \)
41 \( 1 - \)\(64\!\cdots\!42\)\( T + p^{33} T^{2} \)
43 \( 1 + \)\(81\!\cdots\!84\)\( T + p^{33} T^{2} \)
47 \( 1 + \)\(62\!\cdots\!68\)\( T + p^{33} T^{2} \)
53 \( 1 + \)\(21\!\cdots\!94\)\( T + p^{33} T^{2} \)
59 \( 1 - \)\(29\!\cdots\!20\)\( T + p^{33} T^{2} \)
61 \( 1 + \)\(45\!\cdots\!58\)\( T + p^{33} T^{2} \)
67 \( 1 - \)\(11\!\cdots\!12\)\( T + p^{33} T^{2} \)
71 \( 1 - \)\(25\!\cdots\!72\)\( T + p^{33} T^{2} \)
73 \( 1 + \)\(28\!\cdots\!74\)\( T + p^{33} T^{2} \)
79 \( 1 - \)\(92\!\cdots\!20\)\( T + p^{33} T^{2} \)
83 \( 1 + \)\(16\!\cdots\!04\)\( T + p^{33} T^{2} \)
89 \( 1 + \)\(20\!\cdots\!10\)\( T + p^{33} T^{2} \)
97 \( 1 - \)\(22\!\cdots\!42\)\( T + p^{33} T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.29832001039939388914654156345, −17.45589220960381495964401140666, −16.19879606248037124923831154988, −12.97968492005585142239162464054, −10.97703611627622764291287564403, −9.749950439731407749482580089758, −6.55520867329295846541464440379, −5.46997840159020256031697419986, −1.67932471411918471961525408574, 0, 1.67932471411918471961525408574, 5.46997840159020256031697419986, 6.55520867329295846541464440379, 9.749950439731407749482580089758, 10.97703611627622764291287564403, 12.97968492005585142239162464054, 16.19879606248037124923831154988, 17.45589220960381495964401140666, 18.29832001039939388914654156345

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