↑

Properties

Label 2-315-315.292-c1-0-29
Degree $2$
Conductor $315$
Sign $0.950 + 0.310i$
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  + (0.366 + 0.366i)2-s + (1.5 + 0.866i)3-s − 1.73i·4-s + (−1.86 − 1.23i)5-s + (0.232 + 0.866i)6-s + (2 − 1.73i)7-s + (1.36 − 1.36i)8-s + (1.5 + 2.59i)9-s + (−0.232 − 1.13i)10-s + (−0.732 + 1.26i)11-s + (1.49 − 2.59i)12-s + (1 − 0.267i)13-s + (1.36 + 0.0980i)14-s + (−1.73 − 3.46i)15-s − 2.46·16-s + (2.73 + 0.732i)17-s + ⋯
L(s)  = 1  + (0.258 + 0.258i)2-s + (0.866 + 0.499i)3-s − 0.866i·4-s + (−0.834 − 0.550i)5-s + (0.0947 + 0.353i)6-s + (0.755 − 0.654i)7-s + (0.482 − 0.482i)8-s + (0.5 + 0.866i)9-s + (−0.0733 − 0.358i)10-s + (−0.220 + 0.382i)11-s + (0.433 − 0.749i)12-s + (0.277 − 0.0743i)13-s + (0.365 + 0.0262i)14-s + (−0.447 − 0.894i)15-s − 0.616·16-s + (0.662 + 0.177i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.950 + 0.310i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.950 + 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83713 - 0.292393i\)
\(L(\frac12)\) \(\approx\) \(1.83713 - 0.292393i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (1.86 + 1.23i)T \)
7 \( 1 + (-2 + 1.73i)T \)
good2 \( 1 + (-0.366 - 0.366i)T + 2iT^{2} \)
11 \( 1 + (0.732 - 1.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 + 0.267i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.73 - 0.732i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.36 + 5.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.23 + 0.598i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (6 - 3.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + (4.09 - 1.09i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-8.19 - 4.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.59 + 1.5i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.29 - 6.29i)T - 47iT^{2} \)
53 \( 1 + (3.36 + 0.901i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + 9.39iT - 61T^{2} \)
67 \( 1 + (-6.09 - 6.09i)T + 67iT^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (0.169 - 0.633i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 + (-0.366 + 1.36i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.598 - 1.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.09 + 0.830i)T + (84.0 + 48.5i)T^{2} \)
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

−11.36839055636443132413402938537, −10.66763307142068368427589981800, −9.711687966774637749894766769272, −8.773827472163267682415436911727, −7.77660945864385030359266534137, −7.02980844465708049757457324105, −5.15501883649914122438798234700, −4.66356950842849151305800816238, −3.46837213694619530267801223430, −1.43428322760491680100651209789, 2.11917193724551029490627470390, 3.31651718133730233049247331875, 4.07657511510342283526268898989, 5.79312675379174475477129582615, 7.36340518507605661229372967753, 7.909723748541116966177155279148, 8.477991083920513371873725416855, 9.739559648673238345440948015841, 11.20343617835089007713788353557, 11.82338772114712033027653305010

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