Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.227 + 0.973i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.01 − 1.16i)2-s + (−1.21 + 1.23i)3-s + (1.71 + 2.97i)4-s + (0.5 − 0.866i)5-s + (3.89 − 1.07i)6-s + (1.11 − 2.39i)7-s − 3.33i·8-s + (−0.0404 − 2.99i)9-s + (−2.01 + 1.16i)10-s + (2.42 − 1.39i)11-s + (−5.75 − 1.50i)12-s − 3.20i·13-s + (−5.04 + 3.53i)14-s + (0.459 + 1.66i)15-s + (−0.459 + 0.795i)16-s + (−0.440 − 0.763i)17-s + ⋯
L(s)  = 1  + (−1.42 − 0.824i)2-s + (−0.702 + 0.711i)3-s + (0.858 + 1.48i)4-s + (0.223 − 0.387i)5-s + (1.58 − 0.437i)6-s + (0.422 − 0.906i)7-s − 1.18i·8-s + (−0.0134 − 0.999i)9-s + (−0.638 + 0.368i)10-s + (0.729 − 0.421i)11-s + (−1.66 − 0.433i)12-s − 0.888i·13-s + (−1.34 + 0.946i)14-s + (0.118 + 0.431i)15-s + (−0.114 + 0.198i)16-s + (−0.106 − 0.185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.227 + 0.973i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (26, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.227 + 0.973i)$
$L(1)$  $\approx$  $0.357162 - 0.283329i$
$L(\frac12)$  $\approx$  $0.357162 - 0.283329i$
$L(\frac{3}{2})$   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 \( 1 + (1.21 - 1.23i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.11 + 2.39i)T \)
good2 \( 1 + (2.01 + 1.16i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-2.42 + 1.39i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.20iT - 13T^{2} \)
17 \( 1 + (0.440 + 0.763i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.90 - 1.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.53 - 3.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.15iT - 29T^{2} \)
31 \( 1 + (7.62 - 4.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.203 - 0.352i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.55T + 41T^{2} \)
43 \( 1 + 0.118T + 43T^{2} \)
47 \( 1 + (1.31 - 2.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.46 + 3.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.04 - 3.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.7 - 6.17i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.802 - 1.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.25iT - 71T^{2} \)
73 \( 1 + (-0.192 + 0.110i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.56 + 2.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.666T + 83T^{2} \)
89 \( 1 + (0.437 - 0.757i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.37iT - 97T^{2} \)
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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

−13.25198332920463506099148546299, −11.87467397974010327926292399056, −11.18171868532410412064445070943, −10.31778020255620935606129835219, −9.510810936704514405740974871645, −8.494650348796009933501665538266, −7.14335698229331352861591282516, −5.32056021840744485747838014687, −3.58139211027403480765315332422, −1.00903400038061717024357276711, 1.76990431563914865496509141169, 5.29668981685163018397146649296, 6.59157549360708681307352304586, 7.16518846903497013300984959131, 8.556819772962629444442371617431, 9.379909957346089332787866079307, 10.75680932029630261730227354968, 11.60334193393532182014587318765, 12.79406504200886615258345763439, 14.36963016660576444298237501847

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