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Properties

Label 2-975-975.194-c0-0-2
Degree $2$
Conductor $975$
Sign $-0.248 + 0.968i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.69 + 0.550i)2-s + (−0.587 − 0.809i)3-s + (1.76 − 1.27i)4-s + (0.156 + 0.987i)5-s + (1.44 + 1.04i)6-s + (−1.23 + 1.69i)8-s + (−0.309 + 0.951i)9-s + (−0.809 − 1.58i)10-s + (−0.610 − 1.87i)11-s + (−2.06 − 0.672i)12-s + (−0.951 − 0.309i)13-s + (0.707 − 0.707i)15-s + (0.481 − 1.48i)16-s − 1.78i·18-s + (1.53 + 1.53i)20-s + ⋯
L(s)  = 1  + (−1.69 + 0.550i)2-s + (−0.587 − 0.809i)3-s + (1.76 − 1.27i)4-s + (0.156 + 0.987i)5-s + (1.44 + 1.04i)6-s + (−1.23 + 1.69i)8-s + (−0.309 + 0.951i)9-s + (−0.809 − 1.58i)10-s + (−0.610 − 1.87i)11-s + (−2.06 − 0.672i)12-s + (−0.951 − 0.309i)13-s + (0.707 − 0.707i)15-s + (0.481 − 1.48i)16-s − 1.78i·18-s + (1.53 + 1.53i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.248 + 0.968i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.248 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2035619598\)
\(L(\frac12)\) \(\approx\) \(0.2035619598\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (-0.156 - 0.987i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
good2 \( 1 + (1.69 - 0.550i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.280 + 0.863i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.90iT - T^{2} \)
47 \( 1 + (0.533 + 0.734i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (1.59 - 1.16i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-1.04 + 1.44i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.437 + 1.34i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−10.14120704971960403867569439589, −8.935163313616401674859680837122, −8.162956623746413821515360488689, −7.50181047641364700185440178590, −6.84924156192345470065529318868, −6.01800768817550821206924446049, −5.43019578329342889828219460523, −3.06740159473110115369925503875, −2.01884709036703632792668304713, −0.35261130869510859212234362975, 1.52527424760014020740837703021, 2.71850247024525594941647829745, 4.42198660951342935437482151927, 5.01146745058257593573577422807, 6.42553674426170255174359452909, 7.45894264121629236387308551378, 8.139683185052090967550543359723, 9.297523235059337656742857470353, 9.598334986451504523410073494159, 10.09962023426739160940221808069

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