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Properties

Label 2-975-195.164-c1-0-33
Degree $2$
Conductor $975$
Sign $0.0350 + 0.999i$
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 + (1.96 − 4.19i)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 + (7.99 + 0.738i)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 + (0.803 − 1.71i)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 + (1.88 + 0.173i)18-s + (−0.583 − 0.583i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0350 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0350 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.0350 + 0.999i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.0350 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.514463 - 0.496749i\)
\(L(\frac12)\) \(\approx\) \(0.514463 - 0.496749i\)
\(L(\frac{3}{2})\) 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

−9.781370422126459808092434758168, −9.264093139829549951538532338689, −8.437338038279820603146991953428, −7.88103330602900479225006251629, −6.74230755849908049887216768958, −4.99369417843757841084507915475, −4.15555875173688498868359936668, −2.91169513689457428927626004726, −2.45913282862926383032501593544, −0.57996910854805355395498698691, 1.13227834727701292182896363737, 2.17657716887809237065594075278, 4.15238063477276759371706226585, 5.70242032263963836290783252191, 6.23345262723839338744031018090, 6.98662294483779489316938332869, 7.940104734401477689982194627996, 8.132659393749875029318721505996, 9.330424740069444912963069981408, 9.581902389760302104212302585447

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