↑

Properties

Label 2-315-63.41-c1-0-13
Degree $2$
Conductor $315$
Sign $0.954 - 0.297i$
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  + (−1.13 + 0.653i)2-s + (−0.957 + 1.44i)3-s + (−0.145 + 0.252i)4-s + (0.5 − 0.866i)5-s + (0.140 − 2.25i)6-s + (0.709 − 2.54i)7-s − 2.99i·8-s + (−1.16 − 2.76i)9-s + 1.30i·10-s + (4.98 − 2.88i)11-s + (−0.224 − 0.452i)12-s + (3.14 + 1.81i)13-s + (0.861 + 3.34i)14-s + (0.771 + 1.55i)15-s + (1.66 + 2.88i)16-s − 4.92·17-s + ⋯
L(s)  = 1  + (−0.800 + 0.462i)2-s + (−0.552 + 0.833i)3-s + (−0.0729 + 0.126i)4-s + (0.223 − 0.387i)5-s + (0.0574 − 0.922i)6-s + (0.268 − 0.963i)7-s − 1.05i·8-s + (−0.388 − 0.921i)9-s + 0.413i·10-s + (1.50 − 0.868i)11-s + (−0.0649 − 0.130i)12-s + (0.871 + 0.502i)13-s + (0.230 + 0.895i)14-s + (0.199 + 0.400i)15-s + (0.416 + 0.721i)16-s − 1.19·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.954 - 0.297i$
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: \((2,\ 315,\ (\ :1/2),\ 0.954 - 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723554 + 0.110143i\)
\(L(\frac12)\) \(\approx\) \(0.723554 + 0.110143i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.957 - 1.44i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.709 + 2.54i)T \)
good2 \( 1 + (1.13 - 0.653i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-4.98 + 2.88i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.14 - 1.81i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.92T + 17T^{2} \)
19 \( 1 - 3.02iT - 19T^{2} \)
23 \( 1 + (5.93 + 3.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.16 + 4.13i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.18 - 1.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.80T + 37T^{2} \)
41 \( 1 + (-5.57 + 9.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.136 + 0.235i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.88 - 3.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.51iT - 53T^{2} \)
59 \( 1 + (4.33 - 7.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.81 + 4.50i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.65 + 2.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.95iT - 71T^{2} \)
73 \( 1 + 2.88iT - 73T^{2} \)
79 \( 1 + (-5.15 - 8.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.69 - 2.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.68T + 89T^{2} \)
97 \( 1 + (-4.00 + 2.31i)T + (48.5 - 84.0i)T^{2} \)
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

−11.49721398748115055250182668102, −10.59513343284023027887673272033, −9.716796881223103256236248630616, −8.837974010628318936607276285638, −8.270453091904622113096234366477, −6.67097908356908608648641230084, −6.16045876447666488264398028210, −4.25334347222923223824721080179, −3.92602985968267276083356653530, −0.880046503116908451629512115231, 1.38708018806968745466800970739, 2.44607655747780471970736461999, 4.67483282602723214755489822368, 5.96117759446346733735814139061, 6.63265752682217942220036834624, 8.028490024096609613420778566828, 8.874871725240347847592419018549, 9.709425272313470408950968682051, 10.84845951163249021164302666582, 11.52842013175046534049907820536

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