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Properties

Label 2-975-195.44-c1-0-79
Degree $2$
Conductor $975$
Sign $-0.611 - 0.791i$
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  + (1.89 − 1.89i)2-s + (−0.588 − 1.62i)3-s − 5.15i·4-s + (−4.19 − 1.96i)6-s + (0.157 − 0.157i)7-s + (−5.97 − 5.97i)8-s + (−2.30 + 1.91i)9-s + (−0.635 + 0.635i)11-s + (−8.40 + 3.03i)12-s + (−1.55 + 3.25i)13-s − 0.596i·14-s − 12.2·16-s − 4.97i·17-s + (−0.738 + 7.99i)18-s + (−2.54 + 2.54i)19-s + ⋯
L(s)  = 1  + (1.33 − 1.33i)2-s + (−0.339 − 0.940i)3-s − 2.57i·4-s + (−1.71 − 0.803i)6-s + (0.0595 − 0.0595i)7-s + (−2.11 − 2.11i)8-s + (−0.769 + 0.639i)9-s + (−0.191 + 0.191i)11-s + (−2.42 + 0.876i)12-s + (−0.431 + 0.902i)13-s − 0.159i·14-s − 3.06·16-s − 1.20i·17-s + (−0.173 + 1.88i)18-s + (−0.583 + 0.583i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.611 - 0.791i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.611 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.996218 + 2.02978i\)
\(L(\frac12)\) \(\approx\) \(0.996218 + 2.02978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.588 + 1.62i)T \)
5 \( 1 \)
13 \( 1 + (1.55 - 3.25i)T \)
good2 \( 1 + (-1.89 + 1.89i)T - 2iT^{2} \)
7 \( 1 + (-0.157 + 0.157i)T - 7iT^{2} \)
11 \( 1 + (0.635 - 0.635i)T - 11iT^{2} \)
17 \( 1 + 4.97iT - 17T^{2} \)
19 \( 1 + (2.54 - 2.54i)T - 19iT^{2} \)
23 \( 1 + 7.19iT - 23T^{2} \)
29 \( 1 + 0.282iT - 29T^{2} \)
31 \( 1 + (-5.78 + 5.78i)T - 31iT^{2} \)
37 \( 1 + (-6.85 + 6.85i)T - 37iT^{2} \)
41 \( 1 + (7.78 + 7.78i)T + 41iT^{2} \)
43 \( 1 - 4.34T + 43T^{2} \)
47 \( 1 + (-2.40 - 2.40i)T + 47iT^{2} \)
53 \( 1 - 3.68T + 53T^{2} \)
59 \( 1 + (3.97 - 3.97i)T - 59iT^{2} \)
61 \( 1 + 1.19T + 61T^{2} \)
67 \( 1 + (-8.32 - 8.32i)T + 67iT^{2} \)
71 \( 1 + (4.01 + 4.01i)T + 71iT^{2} \)
73 \( 1 + (-3.14 + 3.14i)T - 73iT^{2} \)
79 \( 1 - 7.87T + 79T^{2} \)
83 \( 1 + (1.01 - 1.01i)T - 83iT^{2} \)
89 \( 1 + (-2.87 + 2.87i)T - 89iT^{2} \)
97 \( 1 + (5.12 + 5.12i)T + 97iT^{2} \)
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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

−9.847136306041035411242895356057, −8.872914795580778374998251987496, −7.52305642712209649615513141943, −6.55881306346295365531968370028, −5.84870585056297831325126729834, −4.83916646699398775556913564778, −4.15676100916660628613646272373, −2.64835218273887468137993368000, −2.13091810408586981208151639859, −0.68191267937588871820348404019, 2.92125734672469359967189108892, 3.69344115812921153913065288946, 4.68633240637530016083904754280, 5.28950008995299913990536705747, 6.07866125003799603383851824924, 6.79844100622377166909147601690, 8.024603631306600710038414787187, 8.448394519287862319823142592858, 9.646732425604191997823243974269, 10.62315697303053288885611748103

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