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Properties

Label 2-105-105.89-c1-0-5
Degree $2$
Conductor $105$
Sign $0.888 - 0.458i$
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  + (−0.322 + 0.558i)2-s + (1.15 − 1.28i)3-s + (0.792 + 1.37i)4-s + (−1.60 + 1.56i)5-s + (0.346 + 1.06i)6-s + (2.64 − 0.105i)7-s − 2.31·8-s + (−0.319 − 2.98i)9-s + (−0.354 − 1.39i)10-s + (3.51 − 2.02i)11-s + (2.68 + 0.567i)12-s − 4.21·13-s + (−0.793 + 1.51i)14-s + (0.155 + 3.86i)15-s + (−0.839 + 1.45i)16-s + (−1.88 + 1.08i)17-s + ⋯
L(s)  = 1  + (−0.227 + 0.394i)2-s + (0.668 − 0.743i)3-s + (0.396 + 0.685i)4-s + (−0.716 + 0.697i)5-s + (0.141 + 0.433i)6-s + (0.999 − 0.0397i)7-s − 0.817·8-s + (−0.106 − 0.994i)9-s + (−0.112 − 0.441i)10-s + (1.05 − 0.611i)11-s + (0.774 + 0.163i)12-s − 1.16·13-s + (−0.212 + 0.403i)14-s + (0.0401 + 0.999i)15-s + (−0.209 + 0.363i)16-s + (−0.457 + 0.263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08973 + 0.264813i\)
\(L(\frac12)\) \(\approx\) \(1.08973 + 0.264813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.15 + 1.28i)T \)
5 \( 1 + (1.60 - 1.56i)T \)
7 \( 1 + (-2.64 + 0.105i)T \)
good2 \( 1 + (0.322 - 0.558i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-3.51 + 2.02i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.21T + 13T^{2} \)
17 \( 1 + (1.88 - 1.08i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.87 + 2.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.322 + 0.558i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.16iT - 29T^{2} \)
31 \( 1 + (0.339 - 0.195i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.69 + 2.13i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.27T + 41T^{2} \)
43 \( 1 - 6.54iT - 43T^{2} \)
47 \( 1 + (-6.75 - 3.90i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.60 - 6.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.66 + 9.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.05 - 3.49i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.56 + 4.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.13iT - 71T^{2} \)
73 \( 1 + (-2.61 - 4.53i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.87 - 3.24i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.27iT - 83T^{2} \)
89 \( 1 + (0.447 - 0.774i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.89T + 97T^{2} \)
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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

−14.19106493047755249698332133961, −12.65833014133936476559743458685, −11.82028586181052961820786804040, −11.01668271282049996788700456348, −9.019261600462022577915780087672, −8.165462869424022265458578940905, −7.29552755332362939098006459684, −6.48200420662360326722017274086, −4.00914982824776817880407194082, −2.51811819916583970801788660921, 2.03582324163793411299623236935, 4.15584224566018479051052609533, 5.18034168967108876351841428953, 7.22728426835040558375154034699, 8.542079191391965566741886018650, 9.389295828215246749286941754750, 10.45562488292911895731727163573, 11.53121680357471363805868250754, 12.29558172440223931590185067343, 14.03009498578094390979601494578

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