Properties

Label 2-105-7.4-c1-0-2
Degree $2$
Conductor $105$
Sign $-0.126 - 0.991i$
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  + (1.36 + 2.36i)2-s + (0.5 − 0.866i)3-s + (−2.73 + 4.73i)4-s + (−0.5 − 0.866i)5-s + 2.73·6-s + (0.866 − 2.5i)7-s − 9.46·8-s + (−0.499 − 0.866i)9-s + (1.36 − 2.36i)10-s + (−0.366 + 0.633i)11-s + (2.73 + 4.73i)12-s + 2.26·13-s + (7.09 − 1.36i)14-s − 0.999·15-s + (−7.46 − 12.9i)16-s + (−1.63 + 2.83i)17-s + ⋯
L(s)  = 1  + (0.965 + 1.67i)2-s + (0.288 − 0.499i)3-s + (−1.36 + 2.36i)4-s + (−0.223 − 0.387i)5-s + 1.11·6-s + (0.327 − 0.944i)7-s − 3.34·8-s + (−0.166 − 0.288i)9-s + (0.431 − 0.748i)10-s + (−0.110 + 0.191i)11-s + (0.788 + 1.36i)12-s + 0.629·13-s + (1.89 − 0.365i)14-s − 0.258·15-s + (−1.86 − 3.23i)16-s + (−0.396 + 0.686i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.126 - 0.991i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.126 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00631 + 1.14268i\)
\(L(\frac12)\) \(\approx\) \(1.00631 + 1.14268i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.866 + 2.5i)T \)
good2 \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (0.366 - 0.633i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.26T + 13T^{2} \)
17 \( 1 + (1.63 - 2.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.23 + 3.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.36 - 4.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.19T + 29T^{2} \)
31 \( 1 + (-0.232 + 0.401i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.59 - 2.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.732T + 41T^{2} \)
43 \( 1 - 3.19T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.19 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.0980 + 0.169i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.33 + 12.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.19T + 71T^{2} \)
73 \( 1 + (6.33 - 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.69 - 6.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 + (7.56 + 13.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−14.02256373761212521540129354722, −13.29537856206769085264096762946, −12.68823127309436882759596105375, −11.24446307576954754682755605979, −9.099337314521597948990495949845, −8.089289450486079690680658953257, −7.28263508513117972600711673571, −6.25473397336995953336814554605, −4.82032773151661681924888466043, −3.70199724640457185469268043618, 2.29473201603628499720629466881, 3.54114164783044621516089240010, 4.81683772249474723549343217743, 6.02061208775852973949673648257, 8.555543927013313073554447965102, 9.546495171709696014616340960084, 10.72274749359991799025795908202, 11.34795815118221066487946809908, 12.34647512908053184135045243243, 13.31384564034363972898429625019

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