↑

Properties

Label 2-315-35.33-c1-0-9
Degree $2$
Conductor $315$
Sign $0.655 - 0.755i$
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  + (0.452 + 1.68i)2-s + (−0.912 + 0.526i)4-s + (0.207 − 2.22i)5-s + (1.18 − 2.36i)7-s + (1.16 + 1.16i)8-s + (3.85 − 0.657i)10-s + (1.53 + 2.65i)11-s + (2.01 − 2.01i)13-s + (4.52 + 0.935i)14-s + (−2.49 + 4.32i)16-s + (−1.46 + 5.47i)17-s + (2.95 − 5.11i)19-s + (0.983 + 2.14i)20-s + (−3.79 + 3.79i)22-s + (−2.50 + 0.671i)23-s + ⋯
L(s)  = 1  + (0.319 + 1.19i)2-s + (−0.456 + 0.263i)4-s + (0.0926 − 0.995i)5-s + (0.448 − 0.893i)7-s + (0.413 + 0.413i)8-s + (1.21 − 0.207i)10-s + (0.463 + 0.801i)11-s + (0.560 − 0.560i)13-s + (1.21 + 0.250i)14-s + (−0.624 + 1.08i)16-s + (−0.355 + 1.32i)17-s + (0.677 − 1.17i)19-s + (0.219 + 0.478i)20-s + (−0.809 + 0.809i)22-s + (−0.522 + 0.140i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.655 - 0.755i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.655 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60134 + 0.730290i\)
\(L(\frac12)\) \(\approx\) \(1.60134 + 0.730290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.207 + 2.22i)T \)
7 \( 1 + (-1.18 + 2.36i)T \)
good2 \( 1 + (-0.452 - 1.68i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-1.53 - 2.65i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.01 + 2.01i)T - 13iT^{2} \)
17 \( 1 + (1.46 - 5.47i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.95 + 5.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.50 - 0.671i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 6.43iT - 29T^{2} \)
31 \( 1 + (4.34 - 2.51i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.09 - 11.5i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.20iT - 41T^{2} \)
43 \( 1 + (1.70 + 1.70i)T + 43iT^{2} \)
47 \( 1 + (5.76 - 1.54i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.67 - 6.23i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.844 + 1.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.38 - 3.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.85 + 0.765i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.17T + 71T^{2} \)
73 \( 1 + (-1.72 - 0.462i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.68 + 3.28i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.13 - 7.13i)T - 83iT^{2} \)
89 \( 1 + (-5.56 + 9.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.68 + 3.68i)T + 97iT^{2} \)
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

−11.86163189643371922566882872124, −10.86086995335279833786179264317, −9.801264287556930537838410321691, −8.563764812145004837349505386151, −7.891016983498217671631497346571, −6.92498632121453641266801504360, −5.90996643498440088197657476602, −4.83748515444719932121784041380, −4.08453800386137357186400049827, −1.55416365123141581539915026691, 1.79600302793501009450565916537, 2.96090325513264722613954378154, 3.89732423429801782255032305926, 5.44051957848265118086042839261, 6.57493073307493402150144628265, 7.70628411422641602305869974060, 9.047282494278161657841391362177, 9.868057019933422632570488403830, 11.07660478093804354942002565261, 11.38115557048753373046595555135

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