↑

Properties

Label 2-975-195.164-c1-0-79
Degree $2$
Conductor $975$
Sign $-0.550 - 0.835i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + (−0.483 − 0.483i)2-s + (0.793 − 1.53i)3-s − 1.53i·4-s + (−1.12 + 0.361i)6-s + (−1.24 − 1.24i)7-s + (−1.70 + 1.70i)8-s + (−1.74 − 2.44i)9-s + (−0.387 − 0.387i)11-s + (−2.35 − 1.21i)12-s + (−3.49 + 0.874i)13-s + 1.20i·14-s − 1.41·16-s + 6.75i·17-s + (−0.339 + 2.02i)18-s + (−1.33 − 1.33i)19-s + ⋯
L(s)  = 1  + (−0.342 − 0.342i)2-s + (0.458 − 0.888i)3-s − 0.765i·4-s + (−0.460 + 0.147i)6-s + (−0.470 − 0.470i)7-s + (−0.604 + 0.604i)8-s + (−0.580 − 0.814i)9-s + (−0.116 − 0.116i)11-s + (−0.680 − 0.350i)12-s + (−0.970 + 0.242i)13-s + 0.322i·14-s − 0.352·16-s + 1.63i·17-s + (−0.0800 + 0.477i)18-s + (−0.305 − 0.305i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.550 - 0.835i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.550 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263596 + 0.489246i\)
\(L(\frac12)\) \(\approx\) \(0.263596 + 0.489246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.793 + 1.53i)T \)
5 \( 1 \)
13 \( 1 + (3.49 - 0.874i)T \)
good2 \( 1 + (0.483 + 0.483i)T + 2iT^{2} \)
7 \( 1 + (1.24 + 1.24i)T + 7iT^{2} \)
11 \( 1 + (0.387 + 0.387i)T + 11iT^{2} \)
17 \( 1 - 6.75iT - 17T^{2} \)
19 \( 1 + (1.33 + 1.33i)T + 19iT^{2} \)
23 \( 1 + 1.69iT - 23T^{2} \)
29 \( 1 + 3.85iT - 29T^{2} \)
31 \( 1 + (-4.06 - 4.06i)T + 31iT^{2} \)
37 \( 1 + (2.36 + 2.36i)T + 37iT^{2} \)
41 \( 1 + (-5.72 + 5.72i)T - 41iT^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + (-5.99 + 5.99i)T - 47iT^{2} \)
53 \( 1 + 8.81T + 53T^{2} \)
59 \( 1 + (1.30 + 1.30i)T + 59iT^{2} \)
61 \( 1 + 1.16T + 61T^{2} \)
67 \( 1 + (-4.66 + 4.66i)T - 67iT^{2} \)
71 \( 1 + (0.915 - 0.915i)T - 71iT^{2} \)
73 \( 1 + (8.11 + 8.11i)T + 73iT^{2} \)
79 \( 1 - 3.85T + 79T^{2} \)
83 \( 1 + (-11.9 - 11.9i)T + 83iT^{2} \)
89 \( 1 + (-5.99 - 5.99i)T + 89iT^{2} \)
97 \( 1 + (1.03 - 1.03i)T - 97iT^{2} \)
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

−9.484854098332772034565412608596, −8.662587968259581912270150343767, −7.915935299440694068467034006861, −6.78107089846079590718983210582, −6.31077975389164099879092169594, −5.20069552597994863756988699920, −3.85679070713834910212621223482, −2.58544936592866801552495923903, −1.66345022399378962552753109983, −0.25482318494820087064501669496, 2.62030505950498839338808965475, 3.14075468712303266967459331665, 4.38819337208829720055372037877, 5.21337149423228711532221805109, 6.42938800871011589028374822358, 7.47039338015094507546462393018, 8.046117836710007392849199884941, 9.069644706063828349028760930958, 9.493312522440276478916652487552, 10.19329448854594979028916481276

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