↑

Properties

Label 2-75-5.4-c17-0-31
Degree $2$
Conductor $75$
Sign $0.894 - 0.447i$
Analytic cond. $137.416$
Root an. cond. $11.7224$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 577. i·2-s − 6.56e3i·3-s − 2.02e5·4-s + 3.79e6·6-s − 2.79e7i·7-s − 4.15e7i·8-s − 4.30e7·9-s + 6.58e8·11-s + 1.33e9i·12-s + 4.25e8i·13-s + 1.61e10·14-s − 2.59e9·16-s + 4.12e10i·17-s − 2.48e10i·18-s − 1.11e11·19-s + ⋯
L(s)  = 1  + 1.59i·2-s − 0.577i·3-s − 1.54·4-s + 0.921·6-s − 1.83i·7-s − 0.875i·8-s − 0.333·9-s + 0.925·11-s + 0.893i·12-s + 0.144i·13-s + 2.92·14-s − 0.151·16-s + 1.43i·17-s − 0.532i·18-s − 1.50·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(137.416\)
Root analytic conductor: \(11.7224\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :17/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.741966807\)
\(L(\frac12)\) \(\approx\) \(1.741966807\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 6.56e3iT \)
5 \( 1 \)
good2 \( 1 - 577. iT - 1.31e5T^{2} \)
7 \( 1 + 2.79e7iT - 2.32e14T^{2} \)
11 \( 1 - 6.58e8T + 5.05e17T^{2} \)
13 \( 1 - 4.25e8iT - 8.65e18T^{2} \)
17 \( 1 - 4.12e10iT - 8.27e20T^{2} \)
19 \( 1 + 1.11e11T + 5.48e21T^{2} \)
23 \( 1 - 2.41e11iT - 1.41e23T^{2} \)
29 \( 1 - 9.10e11T + 7.25e24T^{2} \)
31 \( 1 - 6.61e12T + 2.25e25T^{2} \)
37 \( 1 - 2.53e13iT - 4.56e26T^{2} \)
41 \( 1 - 9.96e12T + 2.61e27T^{2} \)
43 \( 1 + 4.20e12iT - 5.87e27T^{2} \)
47 \( 1 - 1.54e14iT - 2.66e28T^{2} \)
53 \( 1 + 7.54e14iT - 2.05e29T^{2} \)
59 \( 1 - 6.28e14T + 1.27e30T^{2} \)
61 \( 1 - 3.95e14T + 2.24e30T^{2} \)
67 \( 1 + 6.34e15iT - 1.10e31T^{2} \)
71 \( 1 - 1.00e16T + 2.96e31T^{2} \)
73 \( 1 - 8.47e14iT - 4.74e31T^{2} \)
79 \( 1 + 4.80e15T + 1.81e32T^{2} \)
83 \( 1 + 6.98e15iT - 4.21e32T^{2} \)
89 \( 1 - 1.08e16T + 1.37e33T^{2} \)
97 \( 1 + 7.39e16iT - 5.95e33T^{2} \)
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

−11.21363888111033513340833508394, −10.00948021599571991739315030198, −8.507064167513247617637617601396, −7.79696586209805442879583375818, −6.63713635146984575277885906610, −6.36952251519013069077370708918, −4.62839892318975316092748273701, −3.79236068392219734810657182216, −1.61574370280769191553131504428, −0.48898421514290813896889757727, 0.75084787068884372385662986881, 2.24497688601746412548691034430, 2.72027590045848428643365145417, 4.03967854164028207115099742068, 5.06258563638704429614454379536, 6.38914359424312875839922674705, 8.639989984492763149984715870217, 9.149290988870098307429665686477, 10.13492216786558690757990778038, 11.31366693094572262957840265256

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