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Properties

Label 2-105-35.13-c3-0-0
Degree $2$
Conductor $105$
Sign $-0.804 - 0.594i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.79 − 1.79i)2-s + (−2.12 + 2.12i)3-s + 1.52i·4-s + (−11.1 − 0.695i)5-s + 7.63i·6-s + (−12.8 − 13.3i)7-s + (17.1 + 17.1i)8-s − 8.99i·9-s + (−21.3 + 18.8i)10-s − 62.0·11-s + (−3.23 − 3.23i)12-s + (−54.1 + 54.1i)13-s + (−47.1 − 0.963i)14-s + (25.1 − 22.1i)15-s + 49.4·16-s + (9.49 + 9.49i)17-s + ⋯
L(s)  = 1  + (0.636 − 0.636i)2-s + (−0.408 + 0.408i)3-s + 0.190i·4-s + (−0.998 − 0.0622i)5-s + 0.519i·6-s + (−0.692 − 0.721i)7-s + (0.757 + 0.757i)8-s − 0.333i·9-s + (−0.674 + 0.595i)10-s − 1.70·11-s + (−0.0777 − 0.0777i)12-s + (−1.15 + 1.15i)13-s + (−0.899 − 0.0183i)14-s + (0.432 − 0.382i)15-s + 0.773·16-s + (0.135 + 0.135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.594i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.804 - 0.594i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.804 - 0.594i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.100761 + 0.305952i\)
\(L(\frac12)\) \(\approx\) \(0.100761 + 0.305952i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.12 - 2.12i)T \)
5 \( 1 + (11.1 + 0.695i)T \)
7 \( 1 + (12.8 + 13.3i)T \)
good2 \( 1 + (-1.79 + 1.79i)T - 8iT^{2} \)
11 \( 1 + 62.0T + 1.33e3T^{2} \)
13 \( 1 + (54.1 - 54.1i)T - 2.19e3iT^{2} \)
17 \( 1 + (-9.49 - 9.49i)T + 4.91e3iT^{2} \)
19 \( 1 - 102.T + 6.85e3T^{2} \)
23 \( 1 + (67.3 + 67.3i)T + 1.21e4iT^{2} \)
29 \( 1 + 41.5iT - 2.43e4T^{2} \)
31 \( 1 + 35.7iT - 2.97e4T^{2} \)
37 \( 1 + (127. - 127. i)T - 5.06e4iT^{2} \)
41 \( 1 + 76.2iT - 6.89e4T^{2} \)
43 \( 1 + (-161. - 161. i)T + 7.95e4iT^{2} \)
47 \( 1 + (33.7 + 33.7i)T + 1.03e5iT^{2} \)
53 \( 1 + (502. + 502. i)T + 1.48e5iT^{2} \)
59 \( 1 - 137.T + 2.05e5T^{2} \)
61 \( 1 - 254. iT - 2.26e5T^{2} \)
67 \( 1 + (522. - 522. i)T - 3.00e5iT^{2} \)
71 \( 1 - 386.T + 3.57e5T^{2} \)
73 \( 1 + (142. - 142. i)T - 3.89e5iT^{2} \)
79 \( 1 + 160. iT - 4.93e5T^{2} \)
83 \( 1 + (227. - 227. i)T - 5.71e5iT^{2} \)
89 \( 1 + 486.T + 7.04e5T^{2} \)
97 \( 1 + (769. + 769. i)T + 9.12e5iT^{2} \)
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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

−13.44200803233427838593640164865, −12.48067733276895248925763726532, −11.75667474114551767783432114784, −10.76509673843688901232912678838, −9.802465279522499427985232325479, −8.013105690978662446029812248111, −7.12308293673661375733523777345, −5.07211774223089639937785655520, −4.13082830220897976977819248666, −2.88383164324906726530277328511, 0.14556154611807419976785304656, 3.02097539759124662852091294284, 5.03089251599218817521234784169, 5.69592075518349209946723073524, 7.25410109601819443771963238360, 7.86339786593064017677816316261, 9.796627827065414217775883934780, 10.78760390612024485389927630896, 12.24893833870822516249146635392, 12.77101105291613387650994349544

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