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Properties

Label 2-105-35.27-c3-0-21
Degree $2$
Conductor $105$
Sign $-0.989 - 0.142i$
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  + (−0.216 − 0.216i)2-s + (−2.12 − 2.12i)3-s − 7.90i·4-s + (−1.17 + 11.1i)5-s + 0.919i·6-s + (−3.57 − 18.1i)7-s + (−3.44 + 3.44i)8-s + 8.99i·9-s + (2.66 − 2.15i)10-s − 48.7·11-s + (−16.7 + 16.7i)12-s + (9.60 + 9.60i)13-s + (−3.16 + 4.71i)14-s + (26.0 − 21.0i)15-s − 61.7·16-s + (−77.7 + 77.7i)17-s + ⋯
L(s)  = 1  + (−0.0766 − 0.0766i)2-s + (−0.408 − 0.408i)3-s − 0.988i·4-s + (−0.105 + 0.994i)5-s + 0.0625i·6-s + (−0.193 − 0.981i)7-s + (−0.152 + 0.152i)8-s + 0.333i·9-s + (0.0843 − 0.0681i)10-s − 1.33·11-s + (−0.403 + 0.403i)12-s + (0.204 + 0.204i)13-s + (−0.0604 + 0.0900i)14-s + (0.448 − 0.362i)15-s − 0.964·16-s + (−1.10 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.989 - 0.142i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.989 - 0.142i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0244078 + 0.341103i\)
\(L(\frac12)\) \(\approx\) \(0.0244078 + 0.341103i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.17 - 11.1i)T \)
7 \( 1 + (3.57 + 18.1i)T \)
good2 \( 1 + (0.216 + 0.216i)T + 8iT^{2} \)
11 \( 1 + 48.7T + 1.33e3T^{2} \)
13 \( 1 + (-9.60 - 9.60i)T + 2.19e3iT^{2} \)
17 \( 1 + (77.7 - 77.7i)T - 4.91e3iT^{2} \)
19 \( 1 + 76.0T + 6.85e3T^{2} \)
23 \( 1 + (-84.8 + 84.8i)T - 1.21e4iT^{2} \)
29 \( 1 + 153. iT - 2.43e4T^{2} \)
31 \( 1 + 194. iT - 2.97e4T^{2} \)
37 \( 1 + (-162. - 162. i)T + 5.06e4iT^{2} \)
41 \( 1 - 147. iT - 6.89e4T^{2} \)
43 \( 1 + (-97.4 + 97.4i)T - 7.95e4iT^{2} \)
47 \( 1 + (-51.3 + 51.3i)T - 1.03e5iT^{2} \)
53 \( 1 + (-430. + 430. i)T - 1.48e5iT^{2} \)
59 \( 1 + 194.T + 2.05e5T^{2} \)
61 \( 1 + 847. iT - 2.26e5T^{2} \)
67 \( 1 + (547. + 547. i)T + 3.00e5iT^{2} \)
71 \( 1 + 688.T + 3.57e5T^{2} \)
73 \( 1 + (-589. - 589. i)T + 3.89e5iT^{2} \)
79 \( 1 - 342. iT - 4.93e5T^{2} \)
83 \( 1 + (214. + 214. i)T + 5.71e5iT^{2} \)
89 \( 1 + 813.T + 7.04e5T^{2} \)
97 \( 1 + (837. - 837. i)T - 9.12e5iT^{2} \)
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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

−13.01804488537860201790784824530, −11.17188322672733362026482431119, −10.75384299824887916963540631787, −9.943630574585926838831578209181, −8.134356925646642116507714618441, −6.82062746956709584569808049323, −6.07940513380448765029792651419, −4.40533345485484371045364305614, −2.28172876903269023223969108248, −0.19120041953101743277665186170, 2.77043060713578051030124114508, 4.53593745172372597961791520180, 5.57763745280131490360332698443, 7.28407503812920725871138105912, 8.637143188470063677558294369302, 9.153163851093088479834480918462, 10.80930100543136693256513215943, 11.89598908432012150926072816279, 12.74441092519405703591970134418, 13.37029371309227679502323611109

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