Properties

Label 2-105-21.5-c1-0-3
Degree $2$
Conductor $105$
Sign $0.444 - 0.895i$
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  + (1.5 + 0.866i)2-s + 1.73i·3-s + (0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−1.49 + 2.59i)6-s + (−2.5 − 0.866i)7-s − 1.73i·8-s − 2.99·9-s + (1.5 − 0.866i)10-s + (3 − 1.73i)11-s + (−1.50 + 0.866i)12-s + 3.46i·13-s + (−3 − 3.46i)14-s + (1.49 + 0.866i)15-s + (2.49 − 4.33i)16-s + (3 + 5.19i)17-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + 0.999i·3-s + (0.250 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.612 + 1.06i)6-s + (−0.944 − 0.327i)7-s − 0.612i·8-s − 0.999·9-s + (0.474 − 0.273i)10-s + (0.904 − 0.522i)11-s + (−0.433 + 0.249i)12-s + 0.960i·13-s + (−0.801 − 0.925i)14-s + (0.387 + 0.223i)15-s + (0.624 − 1.08i)16-s + (0.727 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31693 + 0.817032i\)
\(L(\frac12)\) \(\approx\) \(1.31693 + 0.817032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good2 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6 + 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 0.866i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.73iT - 29T^{2} \)
31 \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (-3 + 1.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−14.16221337083112004545461272380, −13.16528479108188247367854058641, −12.17778866930277049334532663122, −10.71486310466920217838917030012, −9.638832843579580143076439947509, −8.713229594752641254721866506392, −6.64618256560601271719208051527, −5.88733815386282041729798579625, −4.43470446750789562720769364405, −3.61524603998433737121217567054, 2.35185031018607232999486137825, 3.59483606009000309654685246745, 5.53579999814029133993583250022, 6.51566663884681990143475985179, 7.88175014359809323304505485828, 9.371376049804196766070883257826, 10.78827111660785066695676928056, 12.07714230436148351318471699109, 12.47457273420476726697474355545, 13.40234493524462221707469713664

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