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Properties

Label 2-975-195.38-c1-0-70
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: $\chi_{975} (818, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209221 - 0.736491i\)
\(L(\frac12)\) \(\approx\) \(0.209221 - 0.736491i\)
\(L(\frac{3}{2})\) 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

−9.659786646818267585475124606229, −8.492246617222004150216217237459, −7.986207592912142821856459520061, −7.28878050319583071947629426751, −6.60444833712205533714164375509, −5.35348406001457130827422093166, −3.82602611816195384628729825662, −3.26845878069301966098050757989, −2.30607082367421816341189689510, −0.29229296397500435943090621690, 2.11905156452410085811883280282, 2.85887694989544381373834857462, 4.17244198771550371085028472079, 5.22881603792553853535703967460, 5.86369627622208450023938398621, 6.84774640027818673279075231547, 8.178997816600518797006135970841, 8.805852578110759532623589403751, 9.565997493434904350210632175310, 10.32722611567554889268446261694

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