Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.443 + 0.896i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.243 − 0.907i)2-s + (−1.70 + 0.315i)3-s + (0.967 − 0.558i)4-s + (−1.66 − 1.49i)5-s + (0.700 + 1.46i)6-s + (0.0144 − 2.64i)7-s + (−2.07 − 2.07i)8-s + (2.80 − 1.07i)9-s + (−0.949 + 1.87i)10-s + (−0.630 + 0.363i)11-s + (−1.47 + 1.25i)12-s + (−1.44 + 1.44i)13-s + (−2.40 + 0.630i)14-s + (3.30 + 2.01i)15-s + (−0.257 + 0.446i)16-s + (7.09 + 1.90i)17-s + ⋯
L(s)  = 1  + (−0.171 − 0.641i)2-s + (−0.983 + 0.182i)3-s + (0.483 − 0.279i)4-s + (−0.744 − 0.667i)5-s + (0.285 + 0.599i)6-s + (0.00544 − 0.999i)7-s + (−0.732 − 0.732i)8-s + (0.933 − 0.357i)9-s + (−0.300 + 0.592i)10-s + (−0.189 + 0.109i)11-s + (−0.425 + 0.362i)12-s + (−0.400 + 0.400i)13-s + (−0.642 + 0.168i)14-s + (0.853 + 0.520i)15-s + (−0.0644 + 0.111i)16-s + (1.71 + 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.443 + 0.896i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.443 + 0.896i)$
$L(1)$  $\approx$  $0.363555 - 0.585698i$
$L(\frac12)$  $\approx$  $0.363555 - 0.585698i$
$L(\frac{3}{2})$   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 \( 1 + (1.70 - 0.315i)T \)
5 \( 1 + (1.66 + 1.49i)T \)
7 \( 1 + (-0.0144 + 2.64i)T \)
good2 \( 1 + (0.243 + 0.907i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (0.630 - 0.363i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.44 - 1.44i)T - 13iT^{2} \)
17 \( 1 + (-7.09 - 1.90i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.664 + 0.383i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.13 + 0.840i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.07T + 29T^{2} \)
31 \( 1 + (0.209 + 0.363i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.08 + 1.63i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.44iT - 41T^{2} \)
43 \( 1 + (5.15 - 5.15i)T - 43iT^{2} \)
47 \( 1 + (1.82 + 6.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.41 - 5.26i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.807 - 1.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.78 + 8.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.84 + 6.90i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.06iT - 71T^{2} \)
73 \( 1 + (-15.2 - 4.08i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.80 + 3.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.83 - 1.83i)T + 83iT^{2} \)
89 \( 1 + (6.94 - 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.62 - 5.62i)T + 97iT^{2} \)
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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

−12.90044929332650853343742724757, −12.16936230863744191001660297733, −11.33171023282774931833584867002, −10.43870275692123547098094365888, −9.604847608018215725663936450362, −7.75198756655275126346824905877, −6.64617777854957867684395599612, −5.12224310180122211295235063380, −3.75056974945045593408872159921, −1.00705138499228596445757631019, 2.96673474953201300034861081821, 5.23311755990776794226069642557, 6.28863793425181451845064929974, 7.36924185026483891304056156992, 8.201307408361321070437559808222, 9.987965749731870337551717183559, 11.26083812192737934319299360360, 11.91145649119444063932010161618, 12.65408435680132099520764479409, 14.53945015683476302046212488934

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