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Properties

Label 2-105-21.5-c1-0-2
Degree $2$
Conductor $105$
Sign $0.302 - 0.953i$
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  + (2.01 + 1.16i)2-s + (−1.67 + 0.436i)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 + (2.61 − 1.46i)9-s + (−2.01 + 1.16i)10-s + (−2.42 + 1.39i)11-s + (−4.17 − 4.23i)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 + (−0.967 + 0.252i)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 + (0.872 − 0.488i)9-s + (−0.638 + 0.368i)10-s + (−0.729 + 0.421i)11-s + (−1.20 − 1.22i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.302 - 0.953i$
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: \((2,\ 105,\ (\ :1/2),\ 0.302 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24687 + 0.912223i\)
\(L(\frac12)\) \(\approx\) \(1.24687 + 0.912223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 0.436i)T \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.16230159273922943139067231982, −12.90459558025397979219635659217, −12.29816624527268165960954984399, −10.96325426486074253237646573377, −10.19418585171019643782207763811, −7.78414930494540708135101440954, −7.00255773576012537508167509212, −5.75186299474872862797220694619, −4.79068524653722060066706637218, −3.63264221418044970082494965650, 2.11159352161850032204025899971, 4.14718018282436261902934124379, 5.31195287221177057383474190157, 6.01726260523854041776089627658, 7.84011993671429113043375451209, 9.660186657811153917673206405833, 11.17604283530345133212928096279, 11.58799692711307927638176887166, 12.42742784350765086855596112792, 13.27714723860927462004838826730

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