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Properties

Label 2-105-105.38-c2-0-26
Degree $2$
Conductor $105$
Sign $-0.989 - 0.144i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.875 − 3.26i)2-s + (0.286 − 2.98i)3-s + (−6.44 − 3.72i)4-s + (−3.79 + 3.25i)5-s + (−9.50 − 3.55i)6-s + (6.87 − 1.33i)7-s + (−8.24 + 8.24i)8-s + (−8.83 − 1.71i)9-s + (7.32 + 15.2i)10-s + (−1.43 − 0.831i)11-s + (−12.9 + 18.1i)12-s + (15.7 − 15.7i)13-s + (1.65 − 23.6i)14-s + (8.64 + 12.2i)15-s + (4.82 + 8.36i)16-s + (1.94 + 7.27i)17-s + ⋯
L(s)  = 1  + (0.437 − 1.63i)2-s + (0.0954 − 0.995i)3-s + (−1.61 − 0.930i)4-s + (−0.758 + 0.651i)5-s + (−1.58 − 0.591i)6-s + (0.981 − 0.190i)7-s + (−1.03 + 1.03i)8-s + (−0.981 − 0.190i)9-s + (0.732 + 1.52i)10-s + (−0.130 − 0.0755i)11-s + (−1.08 + 1.51i)12-s + (1.21 − 1.21i)13-s + (0.118 − 1.68i)14-s + (0.576 + 0.817i)15-s + (0.301 + 0.522i)16-s + (0.114 + 0.427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.989 - 0.144i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.989 - 0.144i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.106762 + 1.47188i\)
\(L(\frac12)\) \(\approx\) \(0.106762 + 1.47188i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.286 + 2.98i)T \)
5 \( 1 + (3.79 - 3.25i)T \)
7 \( 1 + (-6.87 + 1.33i)T \)
good2 \( 1 + (-0.875 + 3.26i)T + (-3.46 - 2i)T^{2} \)
11 \( 1 + (1.43 + 0.831i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-15.7 + 15.7i)T - 169iT^{2} \)
17 \( 1 + (-1.94 - 7.27i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-6.28 - 10.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (19.2 + 5.15i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 18.4T + 841T^{2} \)
31 \( 1 + (-14.8 - 8.54i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (16.4 + 4.39i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 57.3T + 1.68e3T^{2} \)
43 \( 1 + (-9.66 - 9.66i)T + 1.84e3iT^{2} \)
47 \( 1 + (-86.5 - 23.1i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-5.02 - 18.7i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (65.3 + 37.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (44.4 - 25.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-20.6 - 77.0i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 34.3iT - 5.04e3T^{2} \)
73 \( 1 + (-12.6 - 47.3i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (109. - 63.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (44.3 + 44.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (61.8 - 35.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-73.5 - 73.5i)T + 9.40e3iT^{2} \)
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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

−12.66683736552147733717723531799, −11.96192661581241535488957712543, −11.02315423846914833050780614092, −10.46560631097825101541352733453, −8.509412784599104842801783919004, −7.65215540925992027394848924238, −5.82117045074770748840991310241, −4.01255240794303258679039158053, −2.77038307631817210040201275068, −1.10614035395838571729043214119, 4.05711746400769153848571328026, 4.75756675470438790904145161732, 5.89741653668415358778187002387, 7.51449318877644389697509865194, 8.511503921613352865587758300737, 9.129513577898834448436127554579, 11.04526896789541345257939759378, 11.99244061347765043771405831320, 13.70825715487538435106270553568, 14.21678865488062764473328956511

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