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Properties

Label 2-105-35.13-c1-0-2
Degree $2$
Conductor $105$
Sign $0.887 - 0.461i$
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  + (0.540 − 0.540i)2-s + (−0.707 + 0.707i)3-s + 1.41i·4-s + (1.03 + 1.98i)5-s + 0.763i·6-s + (0.614 − 2.57i)7-s + (1.84 + 1.84i)8-s − 1.00i·9-s + (1.63 + 0.510i)10-s − 3.85·11-s + (−1.00 − 1.00i)12-s + (3.66 − 3.66i)13-s + (−1.05 − 1.72i)14-s + (−2.13 − 0.668i)15-s − 0.839·16-s + (−1.49 − 1.49i)17-s + ⋯
L(s)  = 1  + (0.381 − 0.381i)2-s + (−0.408 + 0.408i)3-s + 0.708i·4-s + (0.463 + 0.886i)5-s + 0.311i·6-s + (0.232 − 0.972i)7-s + (0.652 + 0.652i)8-s − 0.333i·9-s + (0.515 + 0.161i)10-s − 1.16·11-s + (−0.289 − 0.289i)12-s + (1.01 − 1.01i)13-s + (−0.282 − 0.460i)14-s + (−0.550 − 0.172i)15-s − 0.209·16-s + (−0.361 − 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.887 - 0.461i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.887 - 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12765 + 0.275700i\)
\(L(\frac12)\) \(\approx\) \(1.12765 + 0.275700i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 - 1.98i)T \)
7 \( 1 + (-0.614 + 2.57i)T \)
good2 \( 1 + (-0.540 + 0.540i)T - 2iT^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 + (-3.66 + 3.66i)T - 13iT^{2} \)
17 \( 1 + (1.49 + 1.49i)T + 17iT^{2} \)
19 \( 1 + 0.0697T + 19T^{2} \)
23 \( 1 + (0.534 + 0.534i)T + 23iT^{2} \)
29 \( 1 + 2.77iT - 29T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 + (-6.18 + 6.18i)T - 37iT^{2} \)
41 \( 1 - 8.68iT - 41T^{2} \)
43 \( 1 + (2.77 + 2.77i)T + 43iT^{2} \)
47 \( 1 + (5.49 + 5.49i)T + 47iT^{2} \)
53 \( 1 + (-6.13 - 6.13i)T + 53iT^{2} \)
59 \( 1 - 6.97T + 59T^{2} \)
61 \( 1 - 14.3iT - 61T^{2} \)
67 \( 1 + (-0.416 + 0.416i)T - 67iT^{2} \)
71 \( 1 + 8.12T + 71T^{2} \)
73 \( 1 + (9.55 - 9.55i)T - 73iT^{2} \)
79 \( 1 - 9.86iT - 79T^{2} \)
83 \( 1 + (1.63 - 1.63i)T - 83iT^{2} \)
89 \( 1 - 5.05T + 89T^{2} \)
97 \( 1 + (6.85 + 6.85i)T + 97iT^{2} \)
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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

−13.43865956838571910387798041195, −13.15534128800676918033353602750, −11.49068099282408697643337768010, −10.80678710274184437418546267218, −10.06788912102999941156668609599, −8.213993155885694165458740265067, −7.18553995974920017023781643621, −5.64550737301246195819968457645, −4.13993469597125800627486414474, −2.84248837881949346888480767133, 1.78353771708452073245574297165, 4.72765139388627056076311910581, 5.62529891424140337140363991638, 6.50716401673144048303527600928, 8.232001479326197544689929374099, 9.299563693317625440963350656639, 10.58628121915368912004553743488, 11.71745959375170326693700460685, 12.95134636219493797674090555660, 13.51433653860440575523903572973

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