Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.637 + 0.770i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.74 − 2.74i)2-s + (−1.22 − 1.22i)3-s − 11.0i·4-s + (−0.683 + 4.95i)5-s − 6.71·6-s + (1.87 − 1.87i)7-s + (−19.3 − 19.3i)8-s + 2.99i·9-s + (11.7 + 15.4i)10-s + 10.9·11-s + (−13.5 + 13.5i)12-s + (8.10 + 8.10i)13-s − 10.2i·14-s + (6.90 − 5.22i)15-s − 61.7·16-s + (−5.51 + 5.51i)17-s + ⋯
L(s)  = 1  + (1.37 − 1.37i)2-s + (−0.408 − 0.408i)3-s − 2.76i·4-s + (−0.136 + 0.990i)5-s − 1.11·6-s + (0.267 − 0.267i)7-s + (−2.41 − 2.41i)8-s + 0.333i·9-s + (1.17 + 1.54i)10-s + 0.993·11-s + (−1.12 + 1.12i)12-s + (0.623 + 0.623i)13-s − 0.732i·14-s + (0.460 − 0.348i)15-s − 3.85·16-s + (−0.324 + 0.324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.637 + 0.770i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.637 + 0.770i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.981531 - 2.08460i\)
\(L(\frac12)\)  \(\approx\)  \(0.981531 - 2.08460i\)
\(L(2)\)   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 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (0.683 - 4.95i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good2 \( 1 + (-2.74 + 2.74i)T - 4iT^{2} \)
11 \( 1 - 10.9T + 121T^{2} \)
13 \( 1 + (-8.10 - 8.10i)T + 169iT^{2} \)
17 \( 1 + (5.51 - 5.51i)T - 289iT^{2} \)
19 \( 1 + 12.1iT - 361T^{2} \)
23 \( 1 + (-24.3 - 24.3i)T + 529iT^{2} \)
29 \( 1 + 14.8iT - 841T^{2} \)
31 \( 1 + 8.07T + 961T^{2} \)
37 \( 1 + (34.6 - 34.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 32.0T + 1.68e3T^{2} \)
43 \( 1 + (13.0 + 13.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (54.1 - 54.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (-6.76 - 6.76i)T + 2.80e3iT^{2} \)
59 \( 1 - 44.4iT - 3.48e3T^{2} \)
61 \( 1 + 84.4T + 3.72e3T^{2} \)
67 \( 1 + (-0.661 + 0.661i)T - 4.48e3iT^{2} \)
71 \( 1 - 103.T + 5.04e3T^{2} \)
73 \( 1 + (55.1 + 55.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 68.8iT - 6.24e3T^{2} \)
83 \( 1 + (71.4 + 71.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 41.6iT - 7.92e3T^{2} \)
97 \( 1 + (-25.2 + 25.2i)T - 9.40e3iT^{2} \)
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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

−13.19515370456965870853260188566, −11.87173668160928241988635079206, −11.32990907239286613956031119771, −10.63663584107220964364593891742, −9.322432100750769685490491080330, −6.93088423719053082070126545603, −6.04370338523042149828408623791, −4.49636387665808648247144254951, −3.27820334716662310684689179509, −1.59476577704947311700677452057, 3.67723302349111736687802892516, 4.77930441905488120289941901749, 5.66246320305292975518146241346, 6.80046240012490756710408164004, 8.247409592425586461706624375316, 9.051710860406030582635476672285, 11.25375691662554222898711743838, 12.29321532284556433136105607570, 12.90641426323085343682108811234, 14.08746644354911169347699461642

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