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Properties

Label 2-105-35.13-c3-0-2
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: $\chi_{105} (13, \cdot )$
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

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

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