↑

Properties

Label 2-975-195.29-c0-0-0
Degree $2$
Conductor $975$
Sign $0.950 + 0.310i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + 0.999i·12-s − i·13-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)19-s − 0.999·21-s + 0.999i·27-s + (0.866 − 0.499i)28-s + 2·31-s + (−0.499 − 0.866i)36-s + (1.73 − i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + 0.999i·12-s − i·13-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)19-s − 0.999·21-s + 0.999i·27-s + (0.866 − 0.499i)28-s + 2·31-s + (−0.499 − 0.866i)36-s + (1.73 − i)37-s + (0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.950 + 0.310i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.950 + 0.310i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9477353685\)
\(L(\frac12)\) \(\approx\) \(0.9477353685\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + iT \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - 2T + T^{2} \)
37 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
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

−10.31199694799455496799735753498, −9.659556414338113172591965501184, −8.540062061510571298901835882094, −7.61486687703837261264346780795, −6.43905230293228733076702848882, −5.79639563999168833871771457496, −5.12914721769627651736049942001, −4.21483940406040708606916644077, −2.63087902862142898857493052749, −1.21217293355148450759970837390, 1.52103248144036848421848201393, 2.70705884150658762051221679584, 4.32327392488439021590355493003, 4.79055650615339204695599332267, 6.37849423142537571493175652620, 6.70155950459184538073384471000, 7.80075570257638120768927486493, 8.178405054506312168667966082334, 9.433451871263119959890824783466, 10.60469753598795742320014688681

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