↑

Properties

Label 2-675-9.2-c2-0-13
Degree $2$
Conductor $675$
Sign $0.402 + 0.915i$
Analytic cond. $18.3924$
Root an. cond. $4.28863$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.19 − 1.84i)2-s + (4.79 + 8.31i)4-s + (−1.28 + 2.22i)7-s − 20.6i·8-s + (8.51 + 4.91i)11-s + (−6.03 − 10.4i)13-s + (8.19 − 4.73i)14-s + (−18.8 + 32.6i)16-s − 4.28i·17-s − 7.16·19-s + (−18.1 − 31.3i)22-s + (0.442 − 0.255i)23-s + 44.4i·26-s − 24.6·28-s + (−26.4 − 15.2i)29-s + ⋯
L(s)  = 1  + (−1.59 − 0.921i)2-s + (1.19 + 2.07i)4-s + (−0.183 + 0.317i)7-s − 2.57i·8-s + (0.773 + 0.446i)11-s + (−0.463 − 0.803i)13-s + (0.585 − 0.338i)14-s + (−1.17 + 2.04i)16-s − 0.252i·17-s − 0.377·19-s + (−0.823 − 1.42i)22-s + (0.0192 − 0.0110i)23-s + 1.71i·26-s − 0.879·28-s + (−0.912 − 0.526i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ 0.402 + 0.915i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7170096318\)
\(L(\frac12)\) \(\approx\) \(0.7170096318\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (3.19 + 1.84i)T + (2 + 3.46i)T^{2} \)
7 \( 1 + (1.28 - 2.22i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.51 - 4.91i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (6.03 + 10.4i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 4.28iT - 289T^{2} \)
19 \( 1 + 7.16T + 361T^{2} \)
23 \( 1 + (-0.442 + 0.255i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (26.4 + 15.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-9.61 - 16.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 1.31T + 1.36e3T^{2} \)
41 \( 1 + (-29.9 + 17.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-25.9 + 44.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-44.1 - 25.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 86.6iT - 2.80e3T^{2} \)
59 \( 1 + (-91.7 + 52.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (15.6 - 27.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-39.0 - 67.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 72.6iT - 5.04e3T^{2} \)
73 \( 1 + 30.3T + 5.32e3T^{2} \)
79 \( 1 + (-57.6 + 99.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (51.9 + 30.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 71.2iT - 7.92e3T^{2} \)
97 \( 1 + (-63.9 + 110. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−10.10501310769259781794289292983, −9.271625569168715360792841019428, −8.775772816692885654428778318651, −7.69373218999204305985107578008, −7.10049826045977069745170634957, −5.83064464065190390493810878009, −4.18492269247378740118607571413, −2.97457741481334362426241295027, −2.00993563310571080036944171857, −0.64581412225438211822541377312, 0.820253119969082507715094432384, 2.11923787947798185947791026845, 3.98798481024525021673355579485, 5.45381869520405937132606582697, 6.44361812077510683816424832457, 7.00148160303335861357719457912, 7.911515514176010062288955519999, 8.766902387953870037537249546525, 9.413532364387146695144518928723, 10.09730017378568308086639576060

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