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Properties

Label 2-975-39.11-c1-0-48
Degree $2$
Conductor $975$
Sign $0.112 + 0.993i$
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  + (−1.45 − 0.389i)2-s + (1.60 + 0.650i)3-s + (0.232 + 0.133i)4-s + (−2.08 − 1.57i)6-s + (−0.366 − 1.36i)7-s + (1.84 + 1.84i)8-s + (2.15 + 2.08i)9-s + (1.06 − 3.97i)11-s + (0.285 + 0.366i)12-s + (−3.59 − 0.232i)13-s + 2.12i·14-s + (−2.23 − 3.86i)16-s + (2.51 − 4.36i)17-s + (−2.31 − 3.87i)18-s + (−3.73 + i)19-s + ⋯
L(s)  = 1  + (−1.02 − 0.275i)2-s + (0.926 + 0.375i)3-s + (0.116 + 0.0669i)4-s + (−0.849 − 0.641i)6-s + (−0.138 − 0.516i)7-s + (0.652 + 0.652i)8-s + (0.717 + 0.696i)9-s + (0.321 − 1.19i)11-s + (0.0823 + 0.105i)12-s + (−0.997 − 0.0643i)13-s + 0.569i·14-s + (−0.558 − 0.966i)16-s + (0.611 − 1.05i)17-s + (−0.546 − 0.913i)18-s + (−0.856 + 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.112 + 0.993i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.112 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.758294 - 0.677274i\)
\(L(\frac12)\) \(\approx\) \(0.758294 - 0.677274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.60 - 0.650i)T \)
5 \( 1 \)
13 \( 1 + (3.59 + 0.232i)T \)
good2 \( 1 + (1.45 + 0.389i)T + (1.73 + i)T^{2} \)
7 \( 1 + (0.366 + 1.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.06 + 3.97i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.51 + 4.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.73 - i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.20 + 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.46 + 2.46i)T + 31iT^{2} \)
37 \( 1 + (-5.23 - 1.40i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (5.42 + 1.45i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.90 + 1.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.25 + 4.25i)T + 47iT^{2} \)
53 \( 1 + 0.779iT - 53T^{2} \)
59 \( 1 + (-2.90 + 0.779i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.53 + 5.73i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.779 - 2.90i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.901 + 0.901i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (-2.90 + 2.90i)T - 83iT^{2} \)
89 \( 1 + (-2.41 + 9.01i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.63 - 0.437i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.838562762009985646160286874998, −9.042036713143096715632766973571, −8.300341880217884106759520340492, −7.72641318215755765117764862659, −6.75856324738576831996609209533, −5.29241780200359921794597280618, −4.36625694810290112315655648925, −3.22909402523703230732184325291, −2.15458054217037510451298117196, −0.63172328643727117808336218954, 1.44162354644239647773155772527, 2.51236741903870793576087209487, 3.88251860717788913105521053772, 4.80293501489739083025645612416, 6.42861368840404837695552168716, 7.09666262275417218061771615229, 7.86539420161560336158243718518, 8.558854329696913102929094400142, 9.238792888616252154844441299979, 9.914332231596868227121185958995

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