↑

Properties

Label 2-315-63.41-c1-0-12
Degree $2$
Conductor $315$
Sign $0.972 + 0.232i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.334 + 0.193i)2-s + (−0.454 − 1.67i)3-s + (−0.925 + 1.60i)4-s + (0.5 − 0.866i)5-s + (0.475 + 0.471i)6-s + (1.56 + 2.13i)7-s − 1.48i·8-s + (−2.58 + 1.51i)9-s + 0.386i·10-s + (3.67 − 2.12i)11-s + (3.09 + 0.818i)12-s + (3.15 + 1.81i)13-s + (−0.935 − 0.413i)14-s + (−1.67 − 0.442i)15-s + (−1.56 − 2.70i)16-s + 2.70·17-s + ⋯
L(s)  = 1  + (−0.236 + 0.136i)2-s + (−0.262 − 0.964i)3-s + (−0.462 + 0.801i)4-s + (0.223 − 0.387i)5-s + (0.194 + 0.192i)6-s + (0.589 + 0.807i)7-s − 0.526i·8-s + (−0.862 + 0.506i)9-s + 0.122i·10-s + (1.10 − 0.639i)11-s + (0.894 + 0.236i)12-s + (0.873 + 0.504i)13-s + (−0.250 − 0.110i)14-s + (−0.432 − 0.114i)15-s + (−0.390 − 0.676i)16-s + 0.655·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.972 + 0.232i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.972 + 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10860 - 0.130514i\)
\(L(\frac12)\) \(\approx\) \(1.10860 - 0.130514i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.454 + 1.67i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.56 - 2.13i)T \)
good2 \( 1 + (0.334 - 0.193i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.15 - 1.81i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
19 \( 1 + 5.17iT - 19T^{2} \)
23 \( 1 + (-3.29 - 1.90i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.67 + 2.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.652 - 0.376i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 + (0.143 - 0.248i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.08 - 8.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.82 + 8.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.71iT - 53T^{2} \)
59 \( 1 + (4.44 - 7.69i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.5 - 6.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.74 - 8.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + 15.7iT - 73T^{2} \)
79 \( 1 + (1.05 + 1.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.95 - 12.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.28T + 89T^{2} \)
97 \( 1 + (11.7 - 6.78i)T + (48.5 - 84.0i)T^{2} \)
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

−11.83038656305886001610120438934, −11.07129849257139934270257450733, −9.194815685256968873343808150428, −8.787907964096816824984126247457, −7.953697178382691974475908435545, −6.82325825668803278034575825397, −5.86190519445452481616424453049, −4.61755332251426442391361343328, −3.03328545712440595893207274537, −1.26726283097321147564605553089, 1.32147056333589168390718390599, 3.59306204160256105002550453938, 4.57619120917105314735711648297, 5.63035706085357312521005216310, 6.62468136556054947884609619856, 8.155855210302438446371020019623, 9.149060823298540329479416531746, 10.03110073776915742251212514768, 10.58629454010002491050520601409, 11.26980093876446127121742144465

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