Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.321 + 0.947i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.933 − 0.539i)2-s + (0.918 − 1.46i)3-s + (−0.418 − 0.725i)4-s + (0.5 − 0.866i)5-s + (−1.64 + 0.876i)6-s + (−2.47 + 0.929i)7-s + 3.05i·8-s + (−1.31 − 2.69i)9-s + (−0.933 + 0.539i)10-s + (3.84 − 2.21i)11-s + (−1.44 − 0.0513i)12-s − 0.955i·13-s + (2.81 + 0.467i)14-s + (−0.812 − 1.52i)15-s + (0.812 − 1.40i)16-s + (−0.253 − 0.439i)17-s + ⋯
L(s)  = 1  + (−0.660 − 0.381i)2-s + (0.530 − 0.847i)3-s + (−0.209 − 0.362i)4-s + (0.223 − 0.387i)5-s + (−0.673 + 0.357i)6-s + (−0.936 + 0.351i)7-s + 1.08i·8-s + (−0.437 − 0.899i)9-s + (−0.295 + 0.170i)10-s + (1.15 − 0.669i)11-s + (−0.418 − 0.0148i)12-s − 0.265i·13-s + (0.752 + 0.125i)14-s + (−0.209 − 0.394i)15-s + (0.203 − 0.351i)16-s + (−0.0615 − 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.321 + 0.947i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (26, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.321 + 0.947i)$
$L(1)$  $\approx$  $0.460874 - 0.642909i$
$L(\frac12)$  $\approx$  $0.460874 - 0.642909i$
$L(\frac{3}{2})$   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 \( 1 + (-0.918 + 1.46i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.47 - 0.929i)T \)
good2 \( 1 + (0.933 + 0.539i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-3.84 + 2.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.955iT - 13T^{2} \)
17 \( 1 + (0.253 + 0.439i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.41 - 2.54i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.72 - 2.14i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.89iT - 29T^{2} \)
31 \( 1 + (-5.10 + 2.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.76 - 6.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.65T + 41T^{2} \)
43 \( 1 + 0.492T + 43T^{2} \)
47 \( 1 + (-3.32 + 5.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.90 - 4.56i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.81 + 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.399 - 0.230i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.85 - 3.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.90iT - 71T^{2} \)
73 \( 1 + (5.46 - 3.15i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.38 - 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + (3.57 - 6.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.91iT - 97T^{2} \)
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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

−13.54276310184699352785528765672, −12.36510326532031262608097494846, −11.45441903359003323280624762725, −9.875653476015482643410482974126, −9.136111566834719829051436229372, −8.346887379176781872967256200753, −6.73164745494367891985903709583, −5.57320561660966983847811457706, −3.18813773818817911676988456048, −1.26368942323815584386174698163, 3.20195871473636375834775400327, 4.39193814066225622794278350050, 6.52350401469066907492064883978, 7.52778398667160157785931821182, 9.020334721082688693273522109672, 9.508731340857060609371578692032, 10.43929218270282030812331567009, 11.96062480081653022803195373179, 13.29008926190224834471033743900, 14.13455788149178833013909034238

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