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Properties

Label 2-105-15.14-c2-0-9
Degree $2$
Conductor $105$
Sign $0.864 + 0.503i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.57·2-s + (2.50 + 1.65i)3-s + 8.75·4-s + (−0.280 − 4.99i)5-s + (−8.94 − 5.90i)6-s − 2.64i·7-s − 16.9·8-s + (3.53 + 8.27i)9-s + (1.00 + 17.8i)10-s − 2.85i·11-s + (21.9 + 14.4i)12-s − 20.2i·13-s + 9.44i·14-s + (7.54 − 12.9i)15-s + 25.6·16-s + 25.2·17-s + ⋯
L(s)  = 1  − 1.78·2-s + (0.834 + 0.550i)3-s + 2.18·4-s + (−0.0560 − 0.998i)5-s + (−1.49 − 0.983i)6-s − 0.377i·7-s − 2.12·8-s + (0.393 + 0.919i)9-s + (0.100 + 1.78i)10-s − 0.259i·11-s + (1.82 + 1.20i)12-s − 1.55i·13-s + 0.674i·14-s + (0.503 − 0.864i)15-s + 1.60·16-s + 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.864 + 0.503i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.864 + 0.503i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.775756 - 0.209427i\)
\(L(\frac12)\) \(\approx\) \(0.775756 - 0.209427i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.50 - 1.65i)T \)
5 \( 1 + (0.280 + 4.99i)T \)
7 \( 1 + 2.64iT \)
good2 \( 1 + 3.57T + 4T^{2} \)
11 \( 1 + 2.85iT - 121T^{2} \)
13 \( 1 + 20.2iT - 169T^{2} \)
17 \( 1 - 25.2T + 289T^{2} \)
19 \( 1 - 23.2T + 361T^{2} \)
23 \( 1 - 6.18T + 529T^{2} \)
29 \( 1 - 10.2iT - 841T^{2} \)
31 \( 1 + 0.392T + 961T^{2} \)
37 \( 1 + 10.7iT - 1.36e3T^{2} \)
41 \( 1 + 46.2iT - 1.68e3T^{2} \)
43 \( 1 - 35.4iT - 1.84e3T^{2} \)
47 \( 1 + 55.3T + 2.20e3T^{2} \)
53 \( 1 + 19.7T + 2.80e3T^{2} \)
59 \( 1 - 71.2iT - 3.48e3T^{2} \)
61 \( 1 + 58.9T + 3.72e3T^{2} \)
67 \( 1 - 95.3iT - 4.48e3T^{2} \)
71 \( 1 + 100. iT - 5.04e3T^{2} \)
73 \( 1 + 1.93iT - 5.32e3T^{2} \)
79 \( 1 - 40.7T + 6.24e3T^{2} \)
83 \( 1 - 52.6T + 6.88e3T^{2} \)
89 \( 1 - 60.7iT - 7.92e3T^{2} \)
97 \( 1 - 37.6iT - 9.40e3T^{2} \)
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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

−13.43032869950455002869854775223, −12.14668264597312641930250055891, −10.72883531124711092760417625540, −9.919830081742410728297788748004, −9.144801835012459264630010273238, −8.055556311371283621835655279256, −7.60753011416612015556647788003, −5.38810602055988127174204328344, −3.20663671336437183093472885246, −1.10053827108313812108243589927, 1.70830190218925474731425248272, 3.11547723735838430481178149894, 6.43374333987810931536220060271, 7.30805514909097652611698673507, 8.110891772708088367037393188330, 9.399579742398600054789821180316, 9.886858543842781089777604508424, 11.36639610440242875887053917247, 12.09257130031653121024027476666, 13.90685269439389560126579799075

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