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  :  $\chi_{105} (8, \cdot )$
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

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

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