Properties

Label 2-105-21.20-c1-0-10
Degree $2$
Conductor $105$
Sign $-0.654 + 0.755i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·2-s − 1.73i·3-s − 0.999·4-s − 5-s − 2.99·6-s + (2 + 1.73i)7-s − 1.73i·8-s − 2.99·9-s + 1.73i·10-s + 3.46i·11-s + 1.73i·12-s + (2.99 − 3.46i)14-s + 1.73i·15-s − 5·16-s + 6·17-s + 5.19i·18-s + ⋯
L(s)  = 1  − 1.22i·2-s − 0.999i·3-s − 0.499·4-s − 0.447·5-s − 1.22·6-s + (0.755 + 0.654i)7-s − 0.612i·8-s − 0.999·9-s + 0.547i·10-s + 1.04i·11-s + 0.499i·12-s + (0.801 − 0.925i)14-s + 0.447i·15-s − 1.25·16-s + 1.45·17-s + 1.22i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.432379 - 0.946436i\)
\(L(\frac12)\) \(\approx\) \(0.432379 - 0.946436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 + T \)
7 \( 1 + (-2 - 1.73i)T \)
good2 \( 1 + 1.73iT - 2T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.82009214947011327893698412020, −12.26108891185029541558057106982, −11.58250802746344809847045416066, −10.55467498088450060964845520155, −9.150575487050004757995450630884, −7.899835355812656162524348600013, −6.77936700043949431351337836028, −4.96741107106739787117608112172, −3.02106091710520419118583338891, −1.60623861147945947594658943182, 3.63401856002163203666662374263, 5.05964223157965452541194320082, 6.07563044325703369545735779630, 7.78988597368559588077758915153, 8.253625310433564150149755056434, 9.769668520548840763317123178411, 11.02441236922778163076569590742, 11.73201949777095029447336782148, 13.75691866366674629273059850121, 14.44085454038547082654550860093

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