Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.628 - 0.777i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.347 + 0.347i)2-s + (1.72 − 0.176i)3-s + 1.75i·4-s + (−1.16 + 1.90i)5-s + (−0.536 + 0.659i)6-s + (−0.707 − 0.707i)7-s + (−1.30 − 1.30i)8-s + (2.93 − 0.607i)9-s + (−0.256 − 1.06i)10-s − 2.67i·11-s + (0.310 + 3.03i)12-s + (2.14 − 2.14i)13-s + 0.490·14-s + (−1.67 + 3.49i)15-s − 2.61·16-s + (3.26 − 3.26i)17-s + ⋯
L(s)  = 1  + (−0.245 + 0.245i)2-s + (0.994 − 0.101i)3-s + 0.879i·4-s + (−0.522 + 0.852i)5-s + (−0.219 + 0.269i)6-s + (−0.267 − 0.267i)7-s + (−0.461 − 0.461i)8-s + (0.979 − 0.202i)9-s + (−0.0810 − 0.337i)10-s − 0.805i·11-s + (0.0895 + 0.874i)12-s + (0.596 − 0.596i)13-s + 0.131·14-s + (−0.432 + 0.901i)15-s − 0.653·16-s + (0.792 − 0.792i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.628 - 0.777i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (92, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.628 - 0.777i)$
$L(1)$  $\approx$  $0.999394 + 0.477499i$
$L(\frac12)$  $\approx$  $0.999394 + 0.477499i$
$L(\frac{3}{2})$   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 \( 1 + (-1.72 + 0.176i)T \)
5 \( 1 + (1.16 - 1.90i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.347 - 0.347i)T - 2iT^{2} \)
11 \( 1 + 2.67iT - 11T^{2} \)
13 \( 1 + (-2.14 + 2.14i)T - 13iT^{2} \)
17 \( 1 + (-3.26 + 3.26i)T - 17iT^{2} \)
19 \( 1 - 5.24iT - 19T^{2} \)
23 \( 1 + (2.54 + 2.54i)T + 23iT^{2} \)
29 \( 1 + 2.86T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 + (2.14 + 2.14i)T + 37iT^{2} \)
41 \( 1 - 11.5iT - 41T^{2} \)
43 \( 1 + (-0.759 + 0.759i)T - 43iT^{2} \)
47 \( 1 + (7.66 - 7.66i)T - 47iT^{2} \)
53 \( 1 + (-4.43 - 4.43i)T + 53iT^{2} \)
59 \( 1 - 0.159T + 59T^{2} \)
61 \( 1 - 4.72T + 61T^{2} \)
67 \( 1 + (5.41 + 5.41i)T + 67iT^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + (-4.16 + 4.16i)T - 73iT^{2} \)
79 \( 1 - 3.89iT - 79T^{2} \)
83 \( 1 + (-4.03 - 4.03i)T + 83iT^{2} \)
89 \( 1 - 3.95T + 89T^{2} \)
97 \( 1 + (1.86 + 1.86i)T + 97iT^{2} \)
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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

−13.99629929881952685835538515709, −12.99614600259727023697680741029, −11.95612078147161737445790212516, −10.65168348885814778328135152447, −9.424073238928098615720663385559, −8.106865450358884966948463111391, −7.66225813772364676972655848085, −6.37827889489260480932320760699, −3.77809911175282258205806578685, −3.06180810509536809677146384878, 1.81373422795442416444879614727, 3.90661136594612864267408192239, 5.32786212587191406862732330196, 7.08719071962389929539008815808, 8.516019556209459160429086854205, 9.244318123518318236909940846262, 10.14354196050353207333152462487, 11.50414716469142426760442560121, 12.67941598058497583187607301336, 13.63660337820365609171197671895

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