Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.448 + 0.893i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.298 + 0.0799i)2-s + (−1.64 − 0.540i)3-s + (−1.64 − 0.952i)4-s + (−1.56 − 1.59i)5-s + (−0.447 − 0.292i)6-s + (0.951 − 2.46i)7-s + (−0.852 − 0.852i)8-s + (2.41 + 1.77i)9-s + (−0.340 − 0.600i)10-s + (0.660 + 0.381i)11-s + (2.19 + 2.45i)12-s + (−2.27 + 2.27i)13-s + (0.481 − 0.660i)14-s + (1.71 + 3.47i)15-s + (1.71 + 2.97i)16-s + (−1.25 − 4.69i)17-s + ⋯
L(s)  = 1  + (0.210 + 0.0565i)2-s + (−0.950 − 0.312i)3-s + (−0.824 − 0.476i)4-s + (−0.701 − 0.712i)5-s + (−0.182 − 0.119i)6-s + (0.359 − 0.933i)7-s + (−0.301 − 0.301i)8-s + (0.805 + 0.593i)9-s + (−0.107 − 0.189i)10-s + (0.199 + 0.114i)11-s + (0.634 + 0.709i)12-s + (−0.629 + 0.629i)13-s + (0.128 − 0.176i)14-s + (0.443 + 0.896i)15-s + (0.429 + 0.744i)16-s + (−0.305 − 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.448 + 0.893i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.448 + 0.893i)$
$L(1)$  $\approx$  $0.299270 - 0.485100i$
$L(\frac12)$  $\approx$  $0.299270 - 0.485100i$
$L(\frac{3}{2})$   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 \( 1 + (1.64 + 0.540i)T \)
5 \( 1 + (1.56 + 1.59i)T \)
7 \( 1 + (-0.951 + 2.46i)T \)
good2 \( 1 + (-0.298 - 0.0799i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-0.660 - 0.381i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.27 - 2.27i)T - 13iT^{2} \)
17 \( 1 + (1.25 + 4.69i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.41 + 0.818i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.98 + 7.39i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 + (-2.96 + 5.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.915 - 3.41i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.35iT - 41T^{2} \)
43 \( 1 + (-2.69 + 2.69i)T - 43iT^{2} \)
47 \( 1 + (-4.14 - 1.10i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.71 - 1.79i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.84 - 6.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.19 + 3.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0471 - 0.0126i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (-0.359 - 1.34i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.66 - 2.11i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.05 - 5.05i)T + 83iT^{2} \)
89 \( 1 + (-0.453 - 0.785i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.73 - 3.73i)T + 97iT^{2} \)
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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

−13.39693421904091737788949744373, −12.37890719898763375871750840940, −11.49891927439940059660179153478, −10.34240648415674961191168762640, −9.176378494996287975274948437329, −7.75142580882094483501216726538, −6.57780663154673094210617388136, −4.83167292234714006105057090045, −4.47449016277750026230851239256, −0.73766198130993833711533773375, 3.38682358923095611902052775664, 4.74820219836353341832464297619, 5.89695763504139320325289300658, 7.47364299072514358021113402678, 8.676045940345749969920802671268, 9.986606306842582156626500887643, 11.16830411001891080312284504543, 12.06836799151734297787597491878, 12.69692785253600032597887368501, 14.21692182415470514151198545403

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