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Properties

Label 2-105-35.9-c3-0-12
Degree $2$
Conductor $105$
Sign $0.163 - 0.986i$
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  + (4.85 + 2.80i)2-s + (2.59 − 1.5i)3-s + (11.7 + 20.3i)4-s + (−10.6 + 3.37i)5-s + 16.8·6-s + (9.44 − 15.9i)7-s + 86.8i·8-s + (4.5 − 7.79i)9-s + (−61.2 − 13.4i)10-s + (−9.08 − 15.7i)11-s + (61.0 + 35.2i)12-s − 34.8i·13-s + (90.5 − 50.9i)14-s + (−22.6 + 24.7i)15-s + (−149. + 259. i)16-s + (5.05 − 2.91i)17-s + ⋯
L(s)  = 1  + (1.71 + 0.991i)2-s + (0.499 − 0.288i)3-s + (1.46 + 2.54i)4-s + (−0.953 + 0.302i)5-s + 1.14·6-s + (0.509 − 0.860i)7-s + 3.83i·8-s + (0.166 − 0.288i)9-s + (−1.93 − 0.426i)10-s + (−0.248 − 0.431i)11-s + (1.46 + 0.847i)12-s − 0.744i·13-s + (1.72 − 0.972i)14-s + (−0.389 + 0.426i)15-s + (−2.34 + 4.05i)16-s + (0.0721 − 0.0416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.163 - 0.986i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.13109 + 2.65392i\)
\(L(\frac12)\) \(\approx\) \(3.13109 + 2.65392i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.59 + 1.5i)T \)
5 \( 1 + (10.6 - 3.37i)T \)
7 \( 1 + (-9.44 + 15.9i)T \)
good2 \( 1 + (-4.85 - 2.80i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (9.08 + 15.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 34.8iT - 2.19e3T^{2} \)
17 \( 1 + (-5.05 + 2.91i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-30.5 + 52.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-56.5 - 32.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 137.T + 2.43e4T^{2} \)
31 \( 1 + (36.9 + 63.9i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (52.5 + 30.3i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 328.T + 6.89e4T^{2} \)
43 \( 1 - 436. iT - 7.95e4T^{2} \)
47 \( 1 + (-24.8 - 14.3i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-366. + 211. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-220. - 382. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-48.3 + 83.7i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-337. + 194. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 343.T + 3.57e5T^{2} \)
73 \( 1 + (607. - 350. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-40.8 + 70.7i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.35e3iT - 5.71e5T^{2} \)
89 \( 1 + (357. - 619. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 332. iT - 9.12e5T^{2} \)
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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

−13.57057771696311166048614450635, −12.96573303810587538808963749209, −11.73504473793018841939255456091, −10.95460046657246824775639738531, −8.349511816833546262339444187357, −7.60046377641044277324623774601, −6.86135797828958912739284957242, −5.26586325316031093174886284581, −4.00507988820757219659313372229, −3.01751117669771273362266261138, 1.89422593329619346661193028055, 3.37847366750102167797270755102, 4.49839955026452080776646666409, 5.45199590003534633741256513904, 7.16917993236019097740133020500, 8.915617418314147049858543439059, 10.26955257832503160007065480686, 11.42759631182166463404355754848, 12.05603131326193122814817292565, 12.90187703626156058308096964265

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