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Properties

Label 2-975-195.173-c1-0-9
Degree $2$
Conductor $975$
Sign $-0.849 + 0.527i$
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  + (0.561 + 2.09i)2-s + (−1.70 − 0.298i)3-s + (−2.34 + 1.35i)4-s + (−0.333 − 3.74i)6-s + (0.108 − 0.403i)7-s + (−1.09 − 1.09i)8-s + (2.82 + 1.01i)9-s + (−0.255 + 0.443i)11-s + (4.41 − 1.61i)12-s + (0.583 + 3.55i)13-s + 0.907·14-s + (−1.03 + 1.79i)16-s + (−0.493 + 1.84i)17-s + (−0.547 + 6.48i)18-s + (1.50 + 2.60i)19-s + ⋯
L(s)  = 1  + (0.397 + 1.48i)2-s + (−0.985 − 0.172i)3-s + (−1.17 + 0.677i)4-s + (−0.136 − 1.52i)6-s + (0.0408 − 0.152i)7-s + (−0.386 − 0.386i)8-s + (0.940 + 0.339i)9-s + (−0.0771 + 0.133i)11-s + (1.27 − 0.465i)12-s + (0.161 + 0.986i)13-s + 0.242·14-s + (−0.258 + 0.448i)16-s + (−0.119 + 0.446i)17-s + (−0.129 + 1.52i)18-s + (0.345 + 0.597i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.849 + 0.527i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.849 + 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256416 - 0.898832i\)
\(L(\frac12)\) \(\approx\) \(0.256416 - 0.898832i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.298i)T \)
5 \( 1 \)
13 \( 1 + (-0.583 - 3.55i)T \)
good2 \( 1 + (-0.561 - 2.09i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (-0.108 + 0.403i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.255 - 0.443i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.493 - 1.84i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.50 - 2.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.50 + 5.61i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.99 - 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.08iT - 31T^{2} \)
37 \( 1 + (7.16 - 1.92i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.77 - 3.07i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.87 + 6.98i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (5.15 - 5.15i)T - 47iT^{2} \)
53 \( 1 + (7.15 - 7.15i)T - 53iT^{2} \)
59 \( 1 + (1.74 - 1.00i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.39 + 9.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.33 + 2.50i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.93 - 5.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.10 + 2.10i)T - 73iT^{2} \)
79 \( 1 - 14.3iT - 79T^{2} \)
83 \( 1 + (-6.15 - 6.15i)T + 83iT^{2} \)
89 \( 1 + (10.4 + 6.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.19 + 15.6i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.61692072746237911397353565631, −9.525371104021612197113117364207, −8.517311066650077050373895993069, −7.65867727828328951499814021321, −6.86461238640218787600908556574, −6.33772506714529530482736974573, −5.49418464808895285962375185694, −4.69351781458065765912748737706, −3.88753680500341754383487058958, −1.73628878567323535110469036892, 0.43493697310646394020662611379, 1.76092597687058722169542066141, 3.10774262847593537638070239376, 3.96746432209857413819614129068, 5.06249724851503578677561064684, 5.59965799104201255333353430676, 6.84892941774793776518745574947, 7.83263031904549593351854762941, 9.213117606745930344838979571374, 9.857358995336631686856677912967

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