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Properties

Label 2-975-195.77-c1-0-25
Degree $2$
Conductor $975$
Sign $0.850 + 0.525i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s − 2i·4-s + (−2.54 + 2.54i)7-s + 2.99i·9-s + 6.24·11-s + (−2.44 + 2.44i)12-s + (2.54 + 2.54i)13-s − 4·16-s + (1.22 − 1.22i)17-s + 6.24·21-s + (3.67 + 3.67i)23-s + (3.67 − 3.67i)27-s + (5.09 + 5.09i)28-s + (−7.64 − 7.64i)33-s + 5.99·36-s + (−2.54 + 2.54i)37-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s − i·4-s + (−0.963 + 0.963i)7-s + 0.999i·9-s + 1.88·11-s + (−0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s − 16-s + (0.297 − 0.297i)17-s + 1.36·21-s + (0.766 + 0.766i)23-s + (0.707 − 0.707i)27-s + (0.963 + 0.963i)28-s + (−1.33 − 1.33i)33-s + 0.999·36-s + (−0.419 + 0.419i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18468 - 0.336544i\)
\(L(\frac12)\) \(\approx\) \(1.18468 - 0.336544i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
13 \( 1 + (-2.54 - 2.54i)T \)
good2 \( 1 + 2iT^{2} \)
7 \( 1 + (2.54 - 2.54i)T - 7iT^{2} \)
11 \( 1 - 6.24T + 11T^{2} \)
17 \( 1 + (-1.22 + 1.22i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-3.67 - 3.67i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (2.54 - 2.54i)T - 37iT^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (8.57 + 8.57i)T + 53iT^{2} \)
59 \( 1 + 12.4iT - 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + (-10.1 + 10.1i)T - 67iT^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 + (-5.09 - 5.09i)T + 73iT^{2} \)
79 \( 1 - 11iT - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 18.7iT - 89T^{2} \)
97 \( 1 + (2.54 - 2.54i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.584835583098175802077643251546, −9.450825131023342408156559276826, −8.406335501974573585669637394234, −6.82194637746791290833186503748, −6.56907532067074940012122401120, −5.83595109301146870302763581917, −4.96760897164363281234766611929, −3.61231804907073493328102727838, −2.04377637017619597110816430103, −1.02132938634369527862584137591, 0.867348953459630229687356348610, 3.20957995810997093232493963721, 3.81250788114851361248088354429, 4.46319630519698289245566523135, 5.98125416468532587693534961348, 6.63726464907384936657169445215, 7.35128242105188343717980525702, 8.663790656332479888011157129882, 9.237033493946162058511827159819, 10.12700406766687386302767651194

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