Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.971 + 0.238i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.78 − 1.03i)2-s + (−0.627 + 1.61i)3-s + (1.12 − 1.95i)4-s + (0.5 + 0.866i)5-s + (0.543 + 3.53i)6-s + (−0.00953 − 2.64i)7-s − 0.527i·8-s + (−2.21 − 2.02i)9-s + (1.78 + 1.03i)10-s + (−4.06 − 2.34i)11-s + (2.44 + 3.04i)12-s + 0.638i·13-s + (−2.74 − 4.71i)14-s + (−1.71 + 0.263i)15-s + (1.71 + 2.96i)16-s + (−2.07 + 3.59i)17-s + ⋯
L(s)  = 1  + (1.26 − 0.729i)2-s + (−0.362 + 0.932i)3-s + (0.563 − 0.976i)4-s + (0.223 + 0.387i)5-s + (0.221 + 1.44i)6-s + (−0.00360 − 0.999i)7-s − 0.186i·8-s + (−0.737 − 0.675i)9-s + (0.564 + 0.326i)10-s + (−1.22 − 0.707i)11-s + (0.705 + 0.879i)12-s + 0.177i·13-s + (−0.733 − 1.26i)14-s + (−0.442 + 0.0680i)15-s + (0.427 + 0.741i)16-s + (−0.503 + 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.971 + 0.238i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (101, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.971 + 0.238i)$
$L(1)$  $\approx$  $1.60004 - 0.193350i$
$L(\frac12)$  $\approx$  $1.60004 - 0.193350i$
$L(\frac{3}{2})$   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 \( 1 + (0.627 - 1.61i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.00953 + 2.64i)T \)
good2 \( 1 + (-1.78 + 1.03i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (4.06 + 2.34i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.638iT - 13T^{2} \)
17 \( 1 + (2.07 - 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.776 - 0.448i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.89 + 3.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.14iT - 29T^{2} \)
31 \( 1 + (2.02 + 1.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.69 - 9.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.10T + 41T^{2} \)
43 \( 1 - 3.14T + 43T^{2} \)
47 \( 1 + (3.40 + 5.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.96 - 1.13i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.254 - 0.440i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.48 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.41 - 4.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.22iT - 71T^{2} \)
73 \( 1 + (-12.5 - 7.22i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.54 + 7.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.76T + 83T^{2} \)
89 \( 1 + (6.90 + 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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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

−13.56350154359597975818388352208, −12.91067084024626815012074462262, −11.42565350041195789218220765765, −10.78901529827465079229752880765, −10.12794029184922457217913806866, −8.356402065966805306165534145306, −6.42635432795906055589815040963, −5.19942907610908329166890030836, −4.12623637892311151497127713046, −2.93940084751749054223419473568, 2.61578888808864390979676742143, 4.99580413979799486386904975575, 5.54767995398334892871223679998, 6.82937242029342328101228778521, 7.81518613603387369234564069294, 9.303094200776250935996873343872, 11.10690548318724000135432995110, 12.36163389743398264189885353475, 12.86637800941429432627859360542, 13.59057006785725356013798619623

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