Properties

Label 2-105-21.17-c1-0-4
Degree $2$
Conductor $105$
Sign $0.854 + 0.519i$
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  + (−1.78 + 1.03i)2-s + (1.08 − 1.35i)3-s + (1.12 − 1.95i)4-s + (−0.5 − 0.866i)5-s + (−0.543 + 3.53i)6-s + (−0.00953 − 2.64i)7-s + 0.527i·8-s + (−0.649 − 2.92i)9-s + (1.78 + 1.03i)10-s + (4.06 + 2.34i)11-s + (−1.41 − 3.64i)12-s + 0.638i·13-s + (2.74 + 4.71i)14-s + (−1.71 − 0.263i)15-s + (1.71 + 2.96i)16-s + (2.07 − 3.59i)17-s + ⋯
L(s)  = 1  + (−1.26 + 0.729i)2-s + (0.625 − 0.779i)3-s + (0.563 − 0.976i)4-s + (−0.223 − 0.387i)5-s + (−0.221 + 1.44i)6-s + (−0.00360 − 0.999i)7-s + 0.186i·8-s + (−0.216 − 0.976i)9-s + (0.564 + 0.326i)10-s + (1.22 + 0.707i)11-s + (−0.408 − 1.05i)12-s + 0.177i·13-s + (0.733 + 1.26i)14-s + (−0.442 − 0.0680i)15-s + (0.427 + 0.741i)16-s + (0.503 − 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.854 + 0.519i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.854 + 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637814 - 0.178573i\)
\(L(\frac12)\) \(\approx\) \(0.637814 - 0.178573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.08 + 1.35i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.00953 + 2.64i)T \)
good2 \( 1 + (1.78 - 1.03i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-4.06 - 2.34i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.638iT - 13T^{2} \)
17 \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.776 - 0.448i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.89 - 3.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.14iT - 29T^{2} \)
31 \( 1 + (2.02 + 1.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.69 - 9.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.10T + 41T^{2} \)
43 \( 1 - 3.14T + 43T^{2} \)
47 \( 1 + (-3.40 - 5.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.96 + 1.13i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.254 + 0.440i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.48 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.41 - 4.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.22iT - 71T^{2} \)
73 \( 1 + (-12.5 - 7.22i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.54 + 7.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.76T + 83T^{2} \)
89 \( 1 + (-6.90 - 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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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

−13.89369477564430323452959214804, −12.68446911352751571717909750958, −11.58098034022616154159299603027, −9.863942051822707424446078842776, −9.241476197673374214125482119218, −8.007697719196162898322844837019, −7.32873133322106486290608645316, −6.39795124932226386872242763150, −3.96717157370563445646260609250, −1.24971128301304895122118439524, 2.31820321714485861841142038213, 3.78056744349844396015475165600, 5.88824414833555773913366801912, 7.917117547538968072374342393094, 8.740883397409649537344550598561, 9.481025635185043494653202056881, 10.53011941730047780208679524284, 11.37450080145363716579456277611, 12.38149799911079172948426263547, 14.20339448424055159401872550054

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