Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.778 + 0.627i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.260 + 0.260i)2-s + (0.826 − 1.52i)3-s − 1.86i·4-s + (−0.895 + 2.04i)5-s + (0.611 − 0.180i)6-s + (0.707 − 0.707i)7-s + (1.00 − 1.00i)8-s + (−1.63 − 2.51i)9-s + (−0.766 + 0.300i)10-s + 3.38i·11-s + (−2.83 − 1.54i)12-s + (1.59 + 1.59i)13-s + 0.368·14-s + (2.37 + 3.05i)15-s − 3.20·16-s + (0.140 + 0.140i)17-s + ⋯
L(s)  = 1  + (0.184 + 0.184i)2-s + (0.477 − 0.878i)3-s − 0.932i·4-s + (−0.400 + 0.916i)5-s + (0.249 − 0.0738i)6-s + (0.267 − 0.267i)7-s + (0.355 − 0.355i)8-s + (−0.544 − 0.838i)9-s + (−0.242 + 0.0949i)10-s + 1.02i·11-s + (−0.819 − 0.445i)12-s + (0.442 + 0.442i)13-s + 0.0983·14-s + (0.614 + 0.789i)15-s − 0.801·16-s + (0.0341 + 0.0341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.778 + 0.627i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.778 + 0.627i)$
$L(1)$  $\approx$  $1.14359 - 0.403116i$
$L(\frac12)$  $\approx$  $1.14359 - 0.403116i$
$L(\frac{3}{2})$   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 \( 1 + (-0.826 + 1.52i)T \)
5 \( 1 + (0.895 - 2.04i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-0.260 - 0.260i)T + 2iT^{2} \)
11 \( 1 - 3.38iT - 11T^{2} \)
13 \( 1 + (-1.59 - 1.59i)T + 13iT^{2} \)
17 \( 1 + (-0.140 - 0.140i)T + 17iT^{2} \)
19 \( 1 - 7.34iT - 19T^{2} \)
23 \( 1 + (-2.21 + 2.21i)T - 23iT^{2} \)
29 \( 1 + 9.49T + 29T^{2} \)
31 \( 1 - 0.922T + 31T^{2} \)
37 \( 1 + (-5.91 + 5.91i)T - 37iT^{2} \)
41 \( 1 + 1.39iT - 41T^{2} \)
43 \( 1 + (-0.864 - 0.864i)T + 43iT^{2} \)
47 \( 1 + (-0.651 - 0.651i)T + 47iT^{2} \)
53 \( 1 + (-6.54 + 6.54i)T - 53iT^{2} \)
59 \( 1 + 6.25T + 59T^{2} \)
61 \( 1 - 1.83T + 61T^{2} \)
67 \( 1 + (0.815 - 0.815i)T - 67iT^{2} \)
71 \( 1 + 9.77iT - 71T^{2} \)
73 \( 1 + (4.80 + 4.80i)T + 73iT^{2} \)
79 \( 1 - 3.41iT - 79T^{2} \)
83 \( 1 + (6.26 - 6.26i)T - 83iT^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + (6.71 - 6.71i)T - 97iT^{2} \)
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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

−13.93055594181086113782384841961, −12.77498356241661744294250574620, −11.57150451930473504264261123640, −10.52774172827137321804686646642, −9.390747117854820236748114323046, −7.80482914651049242958510038260, −6.95951865767022250172146361519, −5.90815238203584320293055242998, −3.96393331274584351010001409321, −1.92125265782047009007551537095, 3.00840801999611030378276945856, 4.22163758808101973623516964142, 5.38361936595968585756723629624, 7.65850632321200007518909473609, 8.602641212396660019943524803101, 9.217190586820327574897656242623, 11.05212897249473495455999864251, 11.59655886529894241027870391596, 13.09494848569389342272180888526, 13.55223634426232408731711981163

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