↑

Properties

Label 2-391-391.390-c0-0-0
Degree $2$
Conductor $391$
Sign $1$
Analytic cond. $0.195134$
Root an. cond. $0.441740$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.80·2-s + 2.24·4-s − 1.80·5-s − 0.445·7-s − 2.24·8-s + 9-s + 3.24·10-s + 1.24·11-s − 0.445·13-s + 0.801·14-s + 1.80·16-s + 17-s − 1.80·18-s − 4.04·20-s − 2.24·22-s + 23-s + 2.24·25-s + 0.801·26-s − 28-s − 1.00·32-s − 1.80·34-s + 0.801·35-s + 2.24·36-s + 1.24·37-s + 4.04·40-s + 2.80·44-s − 1.80·45-s + ⋯
L(s)  = 1  − 1.80·2-s + 2.24·4-s − 1.80·5-s − 0.445·7-s − 2.24·8-s + 9-s + 3.24·10-s + 1.24·11-s − 0.445·13-s + 0.801·14-s + 1.80·16-s + 17-s − 1.80·18-s − 4.04·20-s − 2.24·22-s + 23-s + 2.24·25-s + 0.801·26-s − 28-s − 1.00·32-s − 1.80·34-s + 0.801·35-s + 2.24·36-s + 1.24·37-s + 4.04·40-s + 2.80·44-s − 1.80·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(391\)    =    \(17 \cdot 23\)
Sign: $1$
Analytic conductor: \(0.195134\)
Root analytic conductor: \(0.441740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{391} (390, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 391,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3098993946\)
\(L(\frac12)\) \(\approx\) \(0.3098993946\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 - T \)
23 \( 1 - T \)
good2 \( 1 + 1.80T + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + 1.80T + T^{2} \)
7 \( 1 + 0.445T + T^{2} \)
11 \( 1 - 1.24T + T^{2} \)
13 \( 1 + 0.445T + T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.24T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.24T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.80T + T^{2} \)
61 \( 1 + 0.445T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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

−11.37514925710046196846582878209, −10.48446805414259649894631543009, −9.547703490173851703373734041208, −8.855826839459769908623998270221, −7.74867286569079367044523927005, −7.33995909206152868120802338742, −6.48572676939444848113121302443, −4.36664393270971181929083928963, −3.19037428923927735858771724883, −1.10737984500835628978997046673, 1.10737984500835628978997046673, 3.19037428923927735858771724883, 4.36664393270971181929083928963, 6.48572676939444848113121302443, 7.33995909206152868120802338742, 7.74867286569079367044523927005, 8.855826839459769908623998270221, 9.547703490173851703373734041208, 10.48446805414259649894631543009, 11.37514925710046196846582878209

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