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Properties

Label 2-105-21.5-c3-0-27
Degree $2$
Conductor $105$
Sign $0.588 + 0.808i$
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  + (1.20 + 0.694i)2-s + (5.16 − 0.577i)3-s + (−3.03 − 5.25i)4-s + (2.5 − 4.33i)5-s + (6.60 + 2.88i)6-s + (−11.1 − 14.8i)7-s − 19.5i·8-s + (26.3 − 5.96i)9-s + (6.01 − 3.47i)10-s + (11.5 − 6.66i)11-s + (−18.7 − 25.4i)12-s + 27.3i·13-s + (−3.09 − 25.5i)14-s + (10.4 − 23.8i)15-s + (−10.7 + 18.5i)16-s + (8.19 + 14.2i)17-s + ⋯
L(s)  = 1  + (0.425 + 0.245i)2-s + (0.993 − 0.111i)3-s + (−0.379 − 0.657i)4-s + (0.223 − 0.387i)5-s + (0.449 + 0.196i)6-s + (−0.600 − 0.799i)7-s − 0.863i·8-s + (0.975 − 0.221i)9-s + (0.190 − 0.109i)10-s + (0.316 − 0.182i)11-s + (−0.450 − 0.611i)12-s + 0.583i·13-s + (−0.0590 − 0.487i)14-s + (0.179 − 0.409i)15-s + (−0.167 + 0.290i)16-s + (0.116 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.588 + 0.808i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.588 + 0.808i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.13583 - 1.08764i\)
\(L(\frac12)\) \(\approx\) \(2.13583 - 1.08764i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.16 + 0.577i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (11.1 + 14.8i)T \)
good2 \( 1 + (-1.20 - 0.694i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-11.5 + 6.66i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 27.3iT - 2.19e3T^{2} \)
17 \( 1 + (-8.19 - 14.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-3.26 - 1.88i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-171. - 98.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 68.2iT - 2.43e4T^{2} \)
31 \( 1 + (4.82 - 2.78i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-125. + 217. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 113.T + 6.89e4T^{2} \)
43 \( 1 - 15.9T + 7.95e4T^{2} \)
47 \( 1 + (199. - 345. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (557. - 321. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-254. - 440. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-315. - 182. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (528. + 914. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 761. iT - 3.57e5T^{2} \)
73 \( 1 + (-399. + 230. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (392. - 680. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + (111. - 192. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.51e3iT - 9.12e5T^{2} \)
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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

−13.38450889017144073832467225859, −12.63563290639166101694408062363, −10.79110039571696593295944689714, −9.581921716837054800259515813976, −9.053137510724016496311322487975, −7.41917395275096911460795450919, −6.33211565128778019246964422428, −4.70009380460893213610123527559, −3.50626865024955336891391358403, −1.24297102879359969813061072566, 2.54736220195490064269870236476, 3.43846149601385358739738828203, 4.98095238655078864286498737065, 6.77477527050953908947009866001, 8.149528143983437272178193830221, 9.038671589356240479303030740277, 10.03493735586132304787105284013, 11.56289984399528511763787260241, 12.78558227890303178087375295392, 13.23563979799129249834038730051

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