↑

Properties

Label 2-975-195.107-c1-0-35
Degree $2$
Conductor $975$
Sign $0.460 + 0.887i$
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  + (−1.67 − 0.448i)3-s + (−1.73 − i)4-s + (3.34 − 0.896i)7-s + (2.59 + 1.50i)9-s + (2.44 + 2.44i)12-s + (−3.60 + 0.0693i)13-s + (1.99 + 3.46i)16-s + (6.06 + 3.5i)19-s − 6·21-s + (−3.67 − 3.67i)27-s + (−6.69 − 1.79i)28-s + 7·31-s + (−3 − 5.19i)36-s + (1.79 − 6.69i)37-s + (6.06 + 1.50i)39-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (−0.866 − 0.5i)4-s + (1.26 − 0.338i)7-s + (0.866 + 0.5i)9-s + (0.707 + 0.707i)12-s + (−0.999 + 0.0192i)13-s + (0.499 + 0.866i)16-s + (1.39 + 0.802i)19-s − 1.30·21-s + (−0.707 − 0.707i)27-s + (−1.26 − 0.338i)28-s + 1.25·31-s + (−0.5 − 0.866i)36-s + (0.294 − 1.10i)37-s + (0.970 + 0.240i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.460 + 0.887i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.460 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.861276 - 0.523473i\)
\(L(\frac12)\) \(\approx\) \(0.861276 - 0.523473i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 + 0.448i)T \)
5 \( 1 \)
13 \( 1 + (3.60 - 0.0693i)T \)
good2 \( 1 + (1.73 + i)T^{2} \)
7 \( 1 + (-3.34 + 0.896i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-6.06 - 3.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + (-1.79 + 6.69i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.68 + 10.0i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.896 - 3.34i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-11.0 + 11.0i)T - 73iT^{2} \)
79 \( 1 + 17iT - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-18.4 + 4.93i)T + (84.0 - 48.5i)T^{2} \)
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

−10.07622819921252800914747926287, −9.165519156453765866854300106766, −7.970277502866107797249258099231, −7.48812597600544789137702408963, −6.28794511555512557804722827927, −5.20081569698496456555368285852, −4.95862767488310638155097861580, −3.89190827364120178174690413436, −1.87546911695623856619500941160, −0.72896683288993166858531455385, 1.05232300382007810037462573541, 2.85753137538482456909586769771, 4.30554060183784985723620453487, 4.92533290492375353418506775927, 5.44868536211168461849652783749, 6.81416262227365588325367437672, 7.72002612345194089310015063523, 8.421340754318157785020096778338, 9.537384226297387170833890093228, 9.955062504670574107948299813271

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