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Properties

Label 2-975-13.9-c1-0-2
Degree $2$
Conductor $975$
Sign $0.352 + 0.935i$
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  + (−1.40 + 2.43i)2-s + (−0.5 + 0.866i)3-s + (−2.95 − 5.11i)4-s + (−1.40 − 2.43i)6-s + (−0.333 − 0.577i)7-s + 10.9·8-s + (−0.499 − 0.866i)9-s + (−1.97 + 3.42i)11-s + 5.90·12-s + (−1.46 + 3.29i)13-s + 1.87·14-s + (−9.54 + 16.5i)16-s + (3.45 + 5.98i)17-s + 2.81·18-s + (−0.656 − 1.13i)19-s + ⋯
L(s)  = 1  + (−0.994 + 1.72i)2-s + (−0.288 + 0.499i)3-s + (−1.47 − 2.55i)4-s + (−0.574 − 0.994i)6-s + (−0.126 − 0.218i)7-s + 3.88·8-s + (−0.166 − 0.288i)9-s + (−0.595 + 1.03i)11-s + 1.70·12-s + (−0.406 + 0.913i)13-s + 0.501·14-s + (−2.38 + 4.13i)16-s + (0.838 + 1.45i)17-s + 0.662·18-s + (−0.150 − 0.260i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.352 + 0.935i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.352 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165792 - 0.114689i\)
\(L(\frac12)\) \(\approx\) \(0.165792 - 0.114689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (1.46 - 3.29i)T \)
good2 \( 1 + (1.40 - 2.43i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (0.333 + 0.577i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.97 - 3.42i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.45 - 5.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.656 + 1.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.800 + 1.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.587 + 1.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.557T + 31T^{2} \)
37 \( 1 + (3.33 - 5.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.45 + 5.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.42 + 5.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.62T + 47T^{2} \)
53 \( 1 + 6.84T + 53T^{2} \)
59 \( 1 + (3.44 + 5.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.34 + 7.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.51 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.0241 - 0.0417i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.72T + 73T^{2} \)
79 \( 1 + 6.02T + 79T^{2} \)
83 \( 1 + 8.60T + 83T^{2} \)
89 \( 1 + (4.22 - 7.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.61 - 2.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.18737725754580532819525828635, −9.799465114252154387374833377979, −8.879517311564579282258228236851, −8.147219679346389840097812766337, −7.25649882260445728919452286618, −6.63973169143383066095120941506, −5.71887067076915890821100985679, −4.89271243590445002610139785182, −4.11787838326435993467619318221, −1.69536777971852305357537395064, 0.15079752628259681488073591073, 1.28237810331585829156934024526, 2.79223646314090858793147133199, 3.14682999104503913643375887483, 4.68146647923250615553324637323, 5.69527356715296106242281243170, 7.39604926885103063235287242269, 7.82972002159597549056746882378, 8.719484689338985255501902655043, 9.512482091393088726192495254516

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