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Properties

Label 2-105-7.4-c3-0-14
Degree $2$
Conductor $105$
Sign $-0.968 - 0.250i$
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  + (−1.5 − 2.59i)2-s + (1.5 − 2.59i)3-s + (−0.5 + 0.866i)4-s + (2.5 + 4.33i)5-s − 9·6-s + (−14 − 12.1i)7-s − 21·8-s + (−4.5 − 7.79i)9-s + (7.50 − 12.9i)10-s + (22.5 − 38.9i)11-s + (1.50 + 2.59i)12-s − 31·13-s + (−10.5 + 54.5i)14-s + 15.0·15-s + (35.5 + 61.4i)16-s + (−48 + 83.1i)17-s + ⋯
L(s)  = 1  + (−0.530 − 0.918i)2-s + (0.288 − 0.499i)3-s + (−0.0625 + 0.108i)4-s + (0.223 + 0.387i)5-s − 0.612·6-s + (−0.755 − 0.654i)7-s − 0.928·8-s + (−0.166 − 0.288i)9-s + (0.237 − 0.410i)10-s + (0.616 − 1.06i)11-s + (0.0360 + 0.0625i)12-s − 0.661·13-s + (−0.200 + 1.04i)14-s + 0.258·15-s + (0.554 + 0.960i)16-s + (−0.684 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.112175 + 0.880264i\)
\(L(\frac12)\) \(\approx\) \(0.112175 + 0.880264i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (14 + 12.1i)T \)
good2 \( 1 + (1.5 + 2.59i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (-22.5 + 38.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 31T + 2.19e3T^{2} \)
17 \( 1 + (48 - 83.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (74.5 + 129. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-70.5 - 122. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 48T + 2.43e4T^{2} \)
31 \( 1 + (-89 + 154. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (185.5 + 321. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 225T + 6.89e4T^{2} \)
43 \( 1 - 344T + 7.95e4T^{2} \)
47 \( 1 + (187.5 + 324. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-331.5 + 574. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-30 + 51.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (196 + 339. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-140 + 242. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 258T + 3.57e5T^{2} \)
73 \( 1 + (289 - 500. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (76 + 131. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 432T + 5.71e5T^{2} \)
89 \( 1 + (-117 - 202. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.35e3T + 9.12e5T^{2} \)
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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

−12.71731659049075508751338423147, −11.36019747000494529343705879662, −10.71666162279910520151234784004, −9.533765159568668018523090237862, −8.697048032929125158960008151639, −6.99942533901203247558686409929, −6.09192577359229907584808315574, −3.63635403752846325746793725004, −2.31274222462139321461689391455, −0.54029290160984290688984810266, 2.68893612810279280526113114914, 4.61135298346137416969617183516, 6.14973915597568207669595150257, 7.16566377084490613404274021757, 8.567695586251581765383851677629, 9.274698754183431621519295825952, 10.15650693903396821863703354695, 12.05098980270461543919130312379, 12.60965301679128368209348384226, 14.25849114318171940177763027898

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