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Properties

Label 2-105-105.104-c1-0-9
Degree $2$
Conductor $105$
Sign $0.897 + 0.441i$
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  + 1.73·2-s + (−1 − 1.41i)3-s + 0.999·4-s + (1.73 + 1.41i)5-s + (−1.73 − 2.44i)6-s + (1 − 2.44i)7-s − 1.73·8-s + (−1.00 + 2.82i)9-s + (2.99 + 2.44i)10-s + 2.82i·11-s + (−0.999 − 1.41i)12-s − 4·13-s + (1.73 − 4.24i)14-s + (0.267 − 3.86i)15-s − 5·16-s + 2.82i·17-s + ⋯
L(s)  = 1  + 1.22·2-s + (−0.577 − 0.816i)3-s + 0.499·4-s + (0.774 + 0.632i)5-s + (−0.707 − 0.999i)6-s + (0.377 − 0.925i)7-s − 0.612·8-s + (−0.333 + 0.942i)9-s + (0.948 + 0.774i)10-s + 0.852i·11-s + (−0.288 − 0.408i)12-s − 1.10·13-s + (0.462 − 1.13i)14-s + (0.0691 − 0.997i)15-s − 1.25·16-s + 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.897 + 0.441i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.897 + 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50069 - 0.348875i\)
\(L(\frac12)\) \(\approx\) \(1.50069 - 0.348875i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1 + 1.41i)T \)
5 \( 1 + (-1.73 - 1.41i)T \)
7 \( 1 + (-1 + 2.44i)T \)
good2 \( 1 - 1.73T + 2T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + 9.79iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 + 4.89iT - 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 9.79iT - 61T^{2} \)
67 \( 1 - 4.89iT - 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−13.54268036931697561545474906643, −12.92923677429546646927667031682, −11.90756009130145738855503962877, −10.84037757618667141998051972768, −9.690602564704510043371776154013, −7.61366997621558420853803955185, −6.68691565206057378813171168210, −5.57778899238268233473338911925, −4.38625364227784555644031732241, −2.34515833163741102245773879332, 3.01381048990394622851526182839, 4.92041555918160653379701026710, 5.22817115143474365843978716757, 6.40265755981170127584780735458, 8.787811672601936250810851613259, 9.480220212427536660710977679283, 10.96473295508290850736651240993, 12.08622520845407308044922596978, 12.65459420231142736873437542713, 14.00393101686152785320184752329

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