Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.266 + 0.963i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−0.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (−0.5 + 0.866i)5-s − 1.99·6-s + (−0.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s + (0.999 + 1.73i)10-s + (3 + 5.19i)11-s + (−1 + 1.73i)12-s − 3·13-s + (−5 − 1.73i)14-s + 0.999·15-s + (1.99 − 3.46i)16-s + (2 + 3.46i)17-s + (0.999 + 1.73i)18-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.288 − 0.499i)3-s + (−0.499 − 0.866i)4-s + (−0.223 + 0.387i)5-s − 0.816·6-s + (−0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (0.904 + 1.56i)11-s + (−0.288 + 0.499i)12-s − 0.832·13-s + (−1.33 − 0.462i)14-s + 0.258·15-s + (0.499 − 0.866i)16-s + (0.485 + 0.840i)17-s + (0.235 + 0.408i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.266 + 0.963i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (16, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.266 + 0.963i)$
$L(1)$  $\approx$  $0.779998 - 1.02529i$
$L(\frac12)$  $\approx$  $0.779998 - 1.02529i$
$L(\frac{3}{2})$   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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−13.06727659824459333800968084157, −12.42797169331384226126533638833, −11.59567067679566165484138677356, −10.50648944710682975799363096614, −9.764649106786253356832991372553, −7.62877944110981313382173921888, −6.74599157926910595495511504158, −4.77609586379963178204813101166, −3.65736222745361496227383056214, −1.83324032084037973742263210554, 3.60095122403794590604848515797, 5.17542032033890415550538412225, 5.81423333869221498209692112067, 7.13040338585399542146241657938, 8.540374159858362232788619976984, 9.458280534738678381554534457946, 11.18868243653864648732214163107, 12.09946190336053656384798194936, 13.31427756038192158740690064611, 14.36217335027837631298114730930

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