↑

Properties

Label 2-738-41.5-c1-0-5
Degree $2$
Conductor $738$
Sign $0.990 - 0.134i$
Analytic cond. $5.89295$
Root an. cond. $2.42754$
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  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (3.17 + 1.03i)5-s + (−0.0295 − 0.186i)7-s + (0.951 − 0.309i)8-s + (−1.03 − 3.17i)10-s + (−1.66 + 3.27i)11-s + (5.44 + 0.861i)13-s + (−0.133 + 0.133i)14-s + (−0.809 − 0.587i)16-s + (−2.15 − 1.09i)17-s + (2.44 − 0.386i)19-s + (−1.96 + 2.70i)20-s + (3.62 − 0.574i)22-s + (−4.08 + 2.96i)23-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.154 + 0.475i)4-s + (1.42 + 0.461i)5-s + (−0.0111 − 0.0704i)7-s + (0.336 − 0.109i)8-s + (−0.326 − 1.00i)10-s + (−0.502 + 0.986i)11-s + (1.50 + 0.239i)13-s + (−0.0356 + 0.0356i)14-s + (−0.202 − 0.146i)16-s + (−0.521 − 0.265i)17-s + (0.560 − 0.0887i)19-s + (−0.439 + 0.604i)20-s + (0.773 − 0.122i)22-s + (−0.850 + 0.618i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $0.990 - 0.134i$
Analytic conductor: \(5.89295\)
Root analytic conductor: \(2.42754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{738} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 738,\ (\ :1/2),\ 0.990 - 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56025 + 0.105675i\)
\(L(\frac12)\) \(\approx\) \(1.56025 + 0.105675i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 \)
41 \( 1 + (-0.312 - 6.39i)T \)
good5 \( 1 + (-3.17 - 1.03i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.0295 + 0.186i)T + (-6.65 + 2.16i)T^{2} \)
11 \( 1 + (1.66 - 3.27i)T + (-6.46 - 8.89i)T^{2} \)
13 \( 1 + (-5.44 - 0.861i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (2.15 + 1.09i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-2.44 + 0.386i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 + (4.08 - 2.96i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.824 - 0.420i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (-0.894 - 2.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.0598 - 0.184i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (0.374 + 0.515i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.339 - 2.14i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-7.80 + 3.97i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (0.834 - 0.606i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-5.01 + 6.90i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (5.92 + 11.6i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (-0.312 + 0.614i)T + (-41.7 - 57.4i)T^{2} \)
73 \( 1 + 6.55iT - 73T^{2} \)
79 \( 1 + (-10.1 - 10.1i)T + 79iT^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + (2.07 + 13.0i)T + (-84.6 + 27.5i)T^{2} \)
97 \( 1 + (2.00 + 3.92i)T + (-57.0 + 78.4i)T^{2} \)
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

−10.31724232528034838380909733048, −9.659681873715886281978654781032, −8.997225216728932947454626498928, −7.917803089672093881292050291652, −6.85715318858780394503688642113, −6.04476918272978058889154696670, −4.99119120935315233296765199138, −3.65815429774190056926882146163, −2.42715828494688572651669687605, −1.53256736846623938751051622739, 1.04518292076479617238225583364, 2.39002259403750628860062518079, 3.97557233225364581364971359969, 5.49867302871101743929920625746, 5.81036958997503643899755829039, 6.64553908553541833571975238165, 8.006516139570301329480844416531, 8.702747880519948875067379218060, 9.291233741474575867827015064742, 10.34907043253566618987032153167

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