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Properties

Label 2-975-65.29-c1-0-14
Degree $2$
Conductor $975$
Sign $-0.260 - 0.965i$
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  + (−2.25 + 1.30i)2-s + (−0.866 + 0.5i)3-s + (2.38 − 4.12i)4-s + (1.30 − 2.25i)6-s + (3.11 + 1.80i)7-s + 7.20i·8-s + (0.499 − 0.866i)9-s + (2.60 + 4.50i)11-s + 4.76i·12-s + (1.97 + 3.01i)13-s − 9.37·14-s + (−4.60 − 7.97i)16-s + (−2.54 − 1.46i)17-s + 2.60i·18-s + (3.38 − 5.86i)19-s + ⋯
L(s)  = 1  + (−1.59 + 0.919i)2-s + (−0.499 + 0.288i)3-s + (1.19 − 2.06i)4-s + (0.531 − 0.919i)6-s + (1.17 + 0.680i)7-s + 2.54i·8-s + (0.166 − 0.288i)9-s + (0.784 + 1.35i)11-s + 1.37i·12-s + (0.547 + 0.837i)13-s − 2.50·14-s + (−1.15 − 1.99i)16-s + (−0.616 − 0.356i)17-s + 0.613i·18-s + (0.776 − 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.260 - 0.965i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.260 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.480904 + 0.627847i\)
\(L(\frac12)\) \(\approx\) \(0.480904 + 0.627847i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-1.97 - 3.01i)T \)
good2 \( 1 + (2.25 - 1.30i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-3.11 - 1.80i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.60 - 4.50i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.54 + 1.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.38 + 5.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.79 + 2.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.916 - 1.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.10T + 31T^{2} \)
37 \( 1 + (-3.06 + 1.76i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.68 - 4.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.74 + 1.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.80iT - 47T^{2} \)
53 \( 1 + 5.20iT - 53T^{2} \)
59 \( 1 + (3.68 - 6.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.71 + 2.97i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.03 - 1.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.85 + 8.40i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.805iT - 73T^{2} \)
79 \( 1 - 4.10T + 79T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 + (-4.91 - 8.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.82 + 2.78i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.941594450106732361439762283632, −9.088762840946097376135086016688, −8.898350791167397802885765176286, −7.76133714784438946781557802446, −6.89576002596222263815371307344, −6.44051945249251614052854487669, −5.13015845536854245566949421429, −4.59002088542521373259621016028, −2.22888272492982428661049423375, −1.14535903776205254175578664103, 0.910380409569910204557259302138, 1.46850725576020344863727632897, 3.06410809706778392471009430256, 4.05720231550303189383574750015, 5.59780976837973387017676608376, 6.65151124470682126728359393370, 7.76418280653478973405334819576, 8.127039665673988904261648480261, 8.907144560400818936508634488110, 9.903872719507961330098262333369

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