Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.494 + 0.868i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.46 − 0.391i)2-s + (−0.852 − 1.50i)3-s + (0.246 − 0.142i)4-s + (0.207 − 2.22i)5-s + (−1.83 − 1.86i)6-s + (2.36 + 1.17i)7-s + (−1.83 + 1.83i)8-s + (−1.54 + 2.57i)9-s + (−0.567 − 3.33i)10-s + (0.791 − 0.457i)11-s + (−0.425 − 0.250i)12-s + (3.07 + 3.07i)13-s + (3.92 + 0.791i)14-s + (−3.53 + 1.58i)15-s + (−2.24 + 3.88i)16-s + (−0.311 + 1.16i)17-s + ⋯
L(s)  = 1  + (1.03 − 0.276i)2-s + (−0.492 − 0.870i)3-s + (0.123 − 0.0712i)4-s + (0.0929 − 0.995i)5-s + (−0.749 − 0.762i)6-s + (0.895 + 0.444i)7-s + (−0.648 + 0.648i)8-s + (−0.514 + 0.857i)9-s + (−0.179 − 1.05i)10-s + (0.238 − 0.137i)11-s + (−0.122 − 0.0723i)12-s + (0.854 + 0.854i)13-s + (1.04 + 0.211i)14-s + (−0.912 + 0.409i)15-s + (−0.561 + 0.971i)16-s + (−0.0755 + 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.494 + 0.868i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (23, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.494 + 0.868i)$
$L(1)$  $\approx$  $1.20792 - 0.702101i$
$L(\frac12)$  $\approx$  $1.20792 - 0.702101i$
$L(\frac{3}{2})$   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 \( 1 + (0.852 + 1.50i)T \)
5 \( 1 + (-0.207 + 2.22i)T \)
7 \( 1 + (-2.36 - 1.17i)T \)
good2 \( 1 + (-1.46 + 0.391i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-0.791 + 0.457i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.07 - 3.07i)T + 13iT^{2} \)
17 \( 1 + (0.311 - 1.16i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.95 + 3.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.505 - 1.88i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.72T + 29T^{2} \)
31 \( 1 + (2.31 + 4.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.207 - 0.774i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.922iT - 41T^{2} \)
43 \( 1 + (4.80 + 4.80i)T + 43iT^{2} \)
47 \( 1 + (10.1 - 2.71i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-10.6 - 2.85i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.94 + 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.533 + 0.924i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.83 - 1.83i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.557iT - 71T^{2} \)
73 \( 1 + (0.564 - 2.10i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.62 + 1.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.38 - 2.38i)T - 83iT^{2} \)
89 \( 1 + (5.64 - 9.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.58 - 1.58i)T - 97iT^{2} \)
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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

−13.37685217783887738515015390753, −12.68660767633995106029380051960, −11.73712095609489215452410432687, −11.15114605802926037586254613799, −8.889221555283975603234675284921, −8.256074449985967076812318599343, −6.38816928945126939974966677221, −5.31037862863709781698238387966, −4.29748543976028554220766308479, −1.94251051800903210010206832643, 3.44771812668954614786566177576, 4.51432204155040616967203390489, 5.74923862391289874919269195721, 6.72204925022024930630197013023, 8.487376690451943609342417553245, 10.07988454190114937277996752450, 10.78815951217563693881898825013, 11.81189390027989181928474375312, 13.11110177819237793458065945660, 14.30272210670142425453303565138

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