Properties

Degree 2
Conductor $ 7 \cdot 157 $
Sign $-1$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 1.41i·3-s − 1.00·4-s + 5-s − 2.00·6-s + 7-s − 1.00·9-s − 1.41i·10-s − 11-s + 1.41i·12-s + 1.41i·13-s − 1.41i·14-s − 1.41i·15-s − 0.999·16-s + 1.41i·17-s + 1.41i·18-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.41i·3-s − 1.00·4-s + 5-s − 2.00·6-s + 7-s − 1.00·9-s − 1.41i·10-s − 11-s + 1.41i·12-s + 1.41i·13-s − 1.41i·14-s − 1.41i·15-s − 0.999·16-s + 1.41i·17-s + 1.41i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1099 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1099 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1099\)    =    \(7 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(0\)
character  :  $\chi_{1099} (1098, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1099,\ (\ :0),\ -1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.255061004\)
\(L(\frac12)\)  \(\approx\)  \(1.255061004\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;157\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{7,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 - T \)
157 \( 1 + T \)
good2 \( 1 + 1.41iT - T^{2} \)
3 \( 1 + 1.41iT - T^{2} \)
5 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.888377561001886859564733883998, −8.808563190495100538701679310691, −8.251207927507114370158544227628, −6.96812051134157432931337368838, −6.50157303852040334319083637926, −5.22311812951748254059338341638, −4.25961002387416376719613084651, −2.60809052647635850507664761521, −2.04019478288948419773736791401, −1.32775542764724275912616004579, 2.28873323938685465760522650434, 3.69409142897207908743715002290, 5.00137227226463542801986455885, 5.41150114349308182059975919530, 5.77791151515662225600748266041, 7.34144124055477092027925249526, 7.904838050419210420809007195296, 8.719103325454428039134100272174, 9.684516409772640015859514790217, 10.14708363041210456982881773240

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