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Properties

Label 2-105-35.27-c1-0-7
Degree $2$
Conductor $105$
Sign $-0.243 + 0.970i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.167 − 0.167i)2-s + (−0.707 − 0.707i)3-s − 1.94i·4-s + (−2.23 − 0.0836i)5-s + 0.236i·6-s + (−0.0627 − 2.64i)7-s + (−0.658 + 0.658i)8-s + 1.00i·9-s + (0.359 + 0.387i)10-s + 3.98·11-s + (−1.37 + 1.37i)12-s + (−0.500 − 0.500i)13-s + (−0.431 + 0.452i)14-s + (1.52 + 1.63i)15-s − 3.66·16-s + (1.67 − 1.67i)17-s + ⋯
L(s)  = 1  + (−0.118 − 0.118i)2-s + (−0.408 − 0.408i)3-s − 0.972i·4-s + (−0.999 − 0.0373i)5-s + 0.0964i·6-s + (−0.0237 − 0.999i)7-s + (−0.232 + 0.232i)8-s + 0.333i·9-s + (0.113 + 0.122i)10-s + 1.20·11-s + (−0.396 + 0.396i)12-s + (−0.138 − 0.138i)13-s + (−0.115 + 0.120i)14-s + (0.392 + 0.423i)15-s − 0.917·16-s + (0.407 − 0.407i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.243 + 0.970i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.243 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.451376 - 0.578430i\)
\(L(\frac12)\) \(\approx\) \(0.451376 - 0.578430i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (2.23 + 0.0836i)T \)
7 \( 1 + (0.0627 + 2.64i)T \)
good2 \( 1 + (0.167 + 0.167i)T + 2iT^{2} \)
11 \( 1 - 3.98T + 11T^{2} \)
13 \( 1 + (0.500 + 0.500i)T + 13iT^{2} \)
17 \( 1 + (-1.67 + 1.67i)T - 17iT^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
23 \( 1 + (5.16 - 5.16i)T - 23iT^{2} \)
29 \( 1 + 3.65iT - 29T^{2} \)
31 \( 1 - 4.93iT - 31T^{2} \)
37 \( 1 + (-0.292 - 0.292i)T + 37iT^{2} \)
41 \( 1 + 7.63iT - 41T^{2} \)
43 \( 1 + (-3.65 + 3.65i)T - 43iT^{2} \)
47 \( 1 + (-0.305 + 0.305i)T - 47iT^{2} \)
53 \( 1 + (-5.39 + 5.39i)T - 53iT^{2} \)
59 \( 1 + 6.10T + 59T^{2} \)
61 \( 1 - 7.11iT - 61T^{2} \)
67 \( 1 + (-0.944 - 0.944i)T + 67iT^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + (1.38 + 1.38i)T + 73iT^{2} \)
79 \( 1 - 8.64iT - 79T^{2} \)
83 \( 1 + (-11.9 - 11.9i)T + 83iT^{2} \)
89 \( 1 + 7.82T + 89T^{2} \)
97 \( 1 + (7.43 - 7.43i)T - 97iT^{2} \)
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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

−13.73045092773093190529177214210, −12.03019222179592167598173248662, −11.51538537082621851102988593721, −10.38127961672762541895265392312, −9.331777830964841595344728815374, −7.69196529923716764615536167086, −6.81180137710398803585408146254, −5.34955478134475027726477575338, −3.84945293073364477529538313597, −1.03403287496494915574867682369, 3.26558424233013049874454371669, 4.46483236407003359209789044905, 6.20902023801625685160530209016, 7.54811009310912085852383990632, 8.613474094958450424204428127201, 9.587028577396067021486546345854, 11.33469557304047772781087134837, 11.96370538890403351525629517681, 12.54424642123591066862001704679, 14.26947133843382326589300635404

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