↑

Properties

Label 2-105-7.4-c3-0-9
Degree $2$
Conductor $105$
Sign $0.491 + 0.871i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.272 − 0.471i)2-s + (−1.5 + 2.59i)3-s + (3.85 − 6.67i)4-s + (2.5 + 4.33i)5-s + 1.63·6-s + (−0.983 − 18.4i)7-s − 8.55·8-s + (−4.5 − 7.79i)9-s + (1.36 − 2.35i)10-s + (−0.182 + 0.316i)11-s + (11.5 + 20.0i)12-s + 72.4·13-s + (−8.45 + 5.49i)14-s − 15.0·15-s + (−28.4 − 49.3i)16-s + (63.8 − 110. i)17-s + ⋯
L(s)  = 1  + (−0.0962 − 0.166i)2-s + (−0.288 + 0.499i)3-s + (0.481 − 0.833i)4-s + (0.223 + 0.387i)5-s + 0.111·6-s + (−0.0530 − 0.998i)7-s − 0.377·8-s + (−0.166 − 0.288i)9-s + (0.0430 − 0.0745i)10-s + (−0.00501 + 0.00868i)11-s + (0.277 + 0.481i)12-s + 1.54·13-s + (−0.161 + 0.104i)14-s − 0.258·15-s + (−0.445 − 0.770i)16-s + (0.911 − 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.871i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.491 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.491 + 0.871i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.491 + 0.871i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.31576 - 0.768673i\)
\(L(\frac12)\) \(\approx\) \(1.31576 - 0.768673i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (0.983 + 18.4i)T \)
good2 \( 1 + (0.272 + 0.471i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (0.182 - 0.316i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 72.4T + 2.19e3T^{2} \)
17 \( 1 + (-63.8 + 110. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (64.7 + 112. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-74.4 - 129. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 21.4T + 2.43e4T^{2} \)
31 \( 1 + (12.7 - 22.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-74.1 - 128. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 213.T + 6.89e4T^{2} \)
43 \( 1 + 285.T + 7.95e4T^{2} \)
47 \( 1 + (-261. - 453. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (109. - 188. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (70.1 - 121. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-237. - 411. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-195. + 338. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 226.T + 3.57e5T^{2} \)
73 \( 1 + (-338. + 586. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (165. + 286. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.27e3T + 5.71e5T^{2} \)
89 \( 1 + (-523. - 907. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.54e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.38621743766484660540006032580, −11.51642997236829549009766170891, −10.98783106292336413746926137536, −10.08825343353021018691413166195, −9.128263545919888230455995427652, −7.24918128190361465553120618413, −6.25206945356235250302910050066, −4.93636741087899934971038082892, −3.19426784280878819957129325815, −0.990804800995632984903878582809, 1.85317445675150874227886538373, 3.65104283835051045912610044816, 5.76902468101081199336276252779, 6.51005342372742580951915797705, 8.294515067329313647867821488272, 8.527961961697769471019903127932, 10.42433728503435046425138994389, 11.58744513022164262281290528144, 12.59245946092813894591633444819, 12.95415604125753536161824397695

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