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Properties

Label 2-975-13.4-c1-0-33
Degree $2$
Conductor $975$
Sign $-0.993 - 0.113i$
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  + (−2.36 − 1.36i)2-s + (0.5 − 0.866i)3-s + (2.72 + 4.71i)4-s + (−2.36 + 1.36i)6-s + (3.23 − 1.86i)7-s − 9.39i·8-s + (−0.499 − 0.866i)9-s + (−2.02 − 1.17i)11-s + 5.44·12-s + (−3.32 + 1.39i)13-s − 10.1·14-s + (−7.36 + 12.7i)16-s + (−1.32 − 2.29i)17-s + 2.72i·18-s + (1.93 − 1.11i)19-s + ⋯
L(s)  = 1  + (−1.67 − 0.964i)2-s + (0.288 − 0.499i)3-s + (1.36 + 2.35i)4-s + (−0.964 + 0.556i)6-s + (1.22 − 0.705i)7-s − 3.32i·8-s + (−0.166 − 0.288i)9-s + (−0.611 − 0.352i)11-s + 1.57·12-s + (−0.922 + 0.386i)13-s − 2.72·14-s + (−1.84 + 3.19i)16-s + (−0.321 − 0.557i)17-s + 0.643i·18-s + (0.444 − 0.256i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.993 - 0.113i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.993 - 0.113i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0326103 + 0.574409i\)
\(L(\frac12)\) \(\approx\) \(0.0326103 + 0.574409i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.32 - 1.39i)T \)
good2 \( 1 + (2.36 + 1.36i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-3.23 + 1.86i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.02 + 1.17i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.32 + 2.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.93 + 1.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.223 - 0.387i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.774 - 1.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.17iT - 31T^{2} \)
37 \( 1 + (1.38 + 0.797i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.866 + 0.500i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.83 + 10.1i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.06iT - 47T^{2} \)
53 \( 1 + 8.33T + 53T^{2} \)
59 \( 1 + (-4.61 + 2.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.317 - 0.550i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.76 - 5.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.28 + 1.89i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.0493iT - 73T^{2} \)
79 \( 1 + 1.93T + 79T^{2} \)
83 \( 1 + 7.63iT - 83T^{2} \)
89 \( 1 + (-4.56 - 2.63i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.8 - 6.24i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.613590932322002641215207972049, −8.723189742487423307828937595046, −8.079958060796109118827734723812, −7.39743513972418006132136646748, −6.88589110002392436302454161044, −5.05019380169471754341459672760, −3.72587371856282986605133126667, −2.53169959224896398901569499638, −1.71658733652005049216427616219, −0.43713427813297040032071715381, 1.60330677713569137644267005731, 2.61108556257510197180730042739, 4.81801388902886328836126521049, 5.34280369085019567000514084406, 6.39014869324155385798652109663, 7.54254703678050711188809921717, 8.039974956479395754765040626005, 8.606831857401737956766058587995, 9.537224175772798996403330581478, 10.05985285661945165582506781891

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