Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.710 + 0.704i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.391 − 1.46i)2-s + (−0.879 − 1.49i)3-s + (−0.246 + 0.142i)4-s + (1.82 − 1.29i)5-s + (−1.83 + 1.86i)6-s + (−1.17 + 2.36i)7-s + (−1.83 − 1.83i)8-s + (−1.45 + 2.62i)9-s + (−2.60 − 2.15i)10-s + (−0.791 + 0.457i)11-s + (0.429 + 0.243i)12-s + (3.07 − 3.07i)13-s + (3.92 + 0.791i)14-s + (−3.53 − 1.58i)15-s + (−2.24 + 3.88i)16-s + (1.16 + 0.311i)17-s + ⋯
L(s)  = 1  + (−0.276 − 1.03i)2-s + (−0.507 − 0.861i)3-s + (−0.123 + 0.0712i)4-s + (0.815 − 0.578i)5-s + (−0.749 + 0.762i)6-s + (−0.444 + 0.895i)7-s + (−0.648 − 0.648i)8-s + (−0.484 + 0.874i)9-s + (−0.822 − 0.682i)10-s + (−0.238 + 0.137i)11-s + (0.124 + 0.0702i)12-s + (0.854 − 0.854i)13-s + (1.04 + 0.211i)14-s + (−0.912 − 0.409i)15-s + (−0.561 + 0.971i)16-s + (0.281 + 0.0755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.710 + 0.704i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.710 + 0.704i)$
$L(1)$  $\approx$  $0.331298 - 0.804514i$
$L(\frac12)$  $\approx$  $0.331298 - 0.804514i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (0.879 + 1.49i)T \)
5 \( 1 + (-1.82 + 1.29i)T \)
7 \( 1 + (1.17 - 2.36i)T \)
good2 \( 1 + (0.391 + 1.46i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (0.791 - 0.457i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.07 + 3.07i)T - 13iT^{2} \)
17 \( 1 + (-1.16 - 0.311i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.95 - 3.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.88 - 0.505i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.72T + 29T^{2} \)
31 \( 1 + (2.31 + 4.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.774 - 0.207i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.922iT - 41T^{2} \)
43 \( 1 + (4.80 - 4.80i)T - 43iT^{2} \)
47 \( 1 + (-2.71 - 10.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.85 - 10.6i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.94 + 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.533 + 0.924i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.83 - 6.83i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.557iT - 71T^{2} \)
73 \( 1 + (-2.10 - 0.564i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.62 - 1.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.38 + 2.38i)T + 83iT^{2} \)
89 \( 1 + (5.64 - 9.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.58 + 1.58i)T + 97iT^{2} \)
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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

−12.86768900673364318847468579194, −12.39553590504107012548461111668, −11.42643746516770855696824855715, −10.26746456174484044721470634627, −9.344510785224519598349934695339, −8.038144938338322050800212712427, −6.22711759228484448306377298645, −5.55437632177330768590485965859, −2.87084453831115097566146032049, −1.38870816652073950795454864894, 3.36232665685655901241319687431, 5.25421233589555496593630846364, 6.38046539980900763529485975758, 7.14220743513215475832518816371, 8.833165568335685518595688298254, 9.828015966412588325421936373741, 10.81741786145503840120653668215, 11.79821094049667137042577187372, 13.61571878366485080399089341272, 14.27465968985452144741059649516

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