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Properties

Label 2-105-35.9-c3-0-14
Degree $2$
Conductor $105$
Sign $-0.760 - 0.648i$
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  + (−4.10 − 2.37i)2-s + (−2.59 + 1.5i)3-s + (7.23 + 12.5i)4-s + (−2.66 − 10.8i)5-s + 14.2·6-s + (15.6 − 9.85i)7-s − 30.7i·8-s + (4.5 − 7.79i)9-s + (−14.8 + 50.8i)10-s + (−18.3 − 31.7i)11-s + (−37.6 − 21.7i)12-s − 11.1i·13-s + (−87.7 + 3.28i)14-s + (23.2 + 24.2i)15-s + (−14.9 + 25.8i)16-s + (−31.6 + 18.2i)17-s + ⋯
L(s)  = 1  + (−1.45 − 0.838i)2-s + (−0.499 + 0.288i)3-s + (0.904 + 1.56i)4-s + (−0.238 − 0.971i)5-s + 0.967·6-s + (0.846 − 0.532i)7-s − 1.35i·8-s + (0.166 − 0.288i)9-s + (−0.468 + 1.60i)10-s + (−0.502 − 0.871i)11-s + (−0.904 − 0.522i)12-s − 0.237i·13-s + (−1.67 + 0.0627i)14-s + (0.399 + 0.416i)15-s + (−0.232 + 0.403i)16-s + (−0.451 + 0.260i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.648i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.760 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.760 - 0.648i$
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: \((2,\ 105,\ (\ :3/2),\ -0.760 - 0.648i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0658604 + 0.178757i\)
\(L(\frac12)\) \(\approx\) \(0.0658604 + 0.178757i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.66 + 10.8i)T \)
7 \( 1 + (-15.6 + 9.85i)T \)
good2 \( 1 + (4.10 + 2.37i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (18.3 + 31.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 11.1iT - 2.19e3T^{2} \)
17 \( 1 + (31.6 - 18.2i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (75.6 - 131. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-59.1 - 34.1i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 167.T + 2.43e4T^{2} \)
31 \( 1 + (130. + 226. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (166. + 95.8i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 142.T + 6.89e4T^{2} \)
43 \( 1 - 122. iT - 7.95e4T^{2} \)
47 \( 1 + (259. + 149. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (362. - 209. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-214. - 371. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (390. - 676. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-794. + 458. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 263.T + 3.57e5T^{2} \)
73 \( 1 + (-276. + 159. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-369. + 639. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 182. iT - 5.71e5T^{2} \)
89 \( 1 + (446. - 772. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 311. iT - 9.12e5T^{2} \)
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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

−12.22076669796850735471423557301, −11.16637836468922823273426659964, −10.66437698545977545725101895869, −9.440021082057126009102905205836, −8.388641684733610130329247248743, −7.69381067590784318331038183218, −5.56261843293285131124500753597, −3.92704982632993507920105541262, −1.61616369707741422053923794402, −0.17994573606956475958737837855, 2.07787777184753756472478721128, 5.03963104866880400130601541325, 6.62277209307107747638187005409, 7.22337535187071123177554322469, 8.332763262532059456444222424122, 9.452001539954740687539891639722, 10.78725941977778345284932889969, 11.22003222481252968838686493676, 12.75080597510435120562988121106, 14.42674438255699019008797703503

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