Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.984 + 0.176i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.631 + 2.35i)2-s + (0.102 + 1.72i)3-s + (−3.42 − 1.97i)4-s + (−1.90 − 1.16i)5-s + (−4.13 − 0.849i)6-s + (1.82 + 1.91i)7-s + (3.36 − 3.36i)8-s + (−2.97 + 0.355i)9-s + (3.94 − 3.76i)10-s + (3.08 + 1.77i)11-s + (3.06 − 6.11i)12-s + (1.28 + 1.28i)13-s + (−5.67 + 3.08i)14-s + (1.81 − 3.42i)15-s + (1.85 + 3.21i)16-s + (−2.95 + 0.792i)17-s + ⋯
L(s)  = 1  + (−0.446 + 1.66i)2-s + (0.0593 + 0.998i)3-s + (−1.71 − 0.987i)4-s + (−0.853 − 0.520i)5-s + (−1.68 − 0.346i)6-s + (0.688 + 0.725i)7-s + (1.19 − 1.19i)8-s + (−0.992 + 0.118i)9-s + (1.24 − 1.18i)10-s + (0.928 + 0.536i)11-s + (0.884 − 1.76i)12-s + (0.356 + 0.356i)13-s + (−1.51 + 0.823i)14-s + (0.469 − 0.883i)15-s + (0.463 + 0.803i)16-s + (−0.717 + 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.984 + 0.176i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (53, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.984 + 0.176i)$
$L(1)$  $\approx$  $0.0616446 - 0.692104i$
$L(\frac12)$  $\approx$  $0.0616446 - 0.692104i$
$L(\frac{3}{2})$   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 \( 1 + (-0.102 - 1.72i)T \)
5 \( 1 + (1.90 + 1.16i)T \)
7 \( 1 + (-1.82 - 1.91i)T \)
good2 \( 1 + (0.631 - 2.35i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-3.08 - 1.77i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.28 - 1.28i)T + 13iT^{2} \)
17 \( 1 + (2.95 - 0.792i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.331 - 0.191i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.45 - 0.658i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.51T + 29T^{2} \)
31 \( 1 + (-0.323 + 0.561i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.00 - 1.34i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (0.335 + 0.335i)T + 43iT^{2} \)
47 \( 1 + (0.751 - 2.80i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.815 - 3.04i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.81 + 6.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.45 + 9.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.31 + 12.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.06iT - 71T^{2} \)
73 \( 1 + (-3.17 + 0.849i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.21 + 1.85i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.973 - 0.973i)T - 83iT^{2} \)
89 \( 1 + (-1.51 - 2.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.3 - 10.3i)T - 97iT^{2} \)
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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

−14.86019868271437175385429770983, −13.92997665715142359974222751698, −12.14526585336271436309381791847, −11.06614915924477481570069087502, −9.363096309525789101589492353067, −8.797793394302929207543971407710, −7.939973568426923581028427226937, −6.48404547334758521761092996063, −5.11058653932666340814068821730, −4.24656157638019536889397932293, 1.04329792440247530466316119725, 2.89990731448436892295095019110, 4.21765563703980365538959664094, 6.70908269094004262879872380372, 8.024826083373493206378317253070, 8.836801804910031763258746822864, 10.47751874907441208167313684006, 11.38214123556977187418393221307, 11.73307404335252538349903607468, 12.96908453551851637475127838032

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