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  :  $\chi_{105} (53, \cdot )$
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

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

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