Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.996 + 0.0873i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.60 + 2.77i)2-s + (−2.69 − 1.32i)3-s + (−3.15 − 5.45i)4-s + (−4.78 + 1.45i)5-s + (7.99 − 5.35i)6-s + (6.02 − 3.55i)7-s + 7.38·8-s + (5.49 + 7.12i)9-s + (3.61 − 15.6i)10-s + (10.5 − 6.09i)11-s + (1.26 + 18.8i)12-s − 8.47i·13-s + (0.220 + 22.4i)14-s + (14.8 + 2.39i)15-s + (0.744 − 1.29i)16-s + (−5.29 − 9.17i)17-s + ⋯
L(s)  = 1  + (−0.802 + 1.38i)2-s + (−0.897 − 0.441i)3-s + (−0.787 − 1.36i)4-s + (−0.956 + 0.291i)5-s + (1.33 − 0.893i)6-s + (0.861 − 0.508i)7-s + 0.923·8-s + (0.610 + 0.791i)9-s + (0.361 − 1.56i)10-s + (0.959 − 0.553i)11-s + (0.105 + 1.57i)12-s − 0.651i·13-s + (0.0157 + 1.60i)14-s + (0.987 + 0.159i)15-s + (0.0465 − 0.0806i)16-s + (−0.311 − 0.539i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.996 + 0.0873i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (44, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.996 + 0.0873i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.491771 - 0.0215280i\)
\(L(\frac12)\)  \(\approx\)  \(0.491771 - 0.0215280i\)
\(L(2)\)   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 \( 1 + (2.69 + 1.32i)T \)
5 \( 1 + (4.78 - 1.45i)T \)
7 \( 1 + (-6.02 + 3.55i)T \)
good2 \( 1 + (1.60 - 2.77i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (-10.5 + 6.09i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 8.47iT - 169T^{2} \)
17 \( 1 + (5.29 + 9.17i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (10.0 - 17.4i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-15.2 + 26.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 42.8iT - 841T^{2} \)
31 \( 1 + (6.11 + 10.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (28.8 + 16.6i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 6.40iT - 1.68e3T^{2} \)
43 \( 1 - 20.0iT - 1.84e3T^{2} \)
47 \( 1 + (-11.8 + 20.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-43.9 - 76.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (41.9 - 24.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-10.4 + 18.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (19.6 - 11.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 2.44iT - 5.04e3T^{2} \)
73 \( 1 + (-76.5 + 44.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (3.20 - 5.54i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 103.T + 6.88e3T^{2} \)
89 \( 1 + (54.2 + 31.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 140. iT - 9.40e3T^{2} \)
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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

−13.89439168384238883970868494795, −12.27389013219697780439830782826, −11.28937090545467403661742589743, −10.35140767200621292432602772150, −8.626710920325621649760875827130, −7.77036899523037491869822771435, −6.94926687354583238073807707712, −5.89875512605958566997437547016, −4.42246698905313280908238081137, −0.60341931898030753111712855625, 1.45100669491318578020145093475, 3.76851446483748331733494666849, 4.90798824278793577160553636355, 6.96941575731355666252195426650, 8.663241851827830389616498200564, 9.292145382830567885484375789163, 10.72691939635638843647716668602, 11.40127354415654204473682633114, 11.96646611653701851401629201220, 12.78289834520279837000126035591

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