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Properties

Label 2-975-195.83-c0-0-0
Degree $2$
Conductor $975$
Sign $0.661 - 0.749i$
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  + (−0.707 − 0.707i)3-s − 4-s − 1.41·7-s + 1.00i·9-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s + 16-s + (1 + i)19-s + (1.00 + 1.00i)21-s + (0.707 − 0.707i)27-s + 1.41·28-s + (−1 + i)31-s − 1.00i·36-s + 1.41·37-s − 1.00i·39-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s − 4-s − 1.41·7-s + 1.00i·9-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s + 16-s + (1 + i)19-s + (1.00 + 1.00i)21-s + (0.707 − 0.707i)27-s + 1.41·28-s + (−1 + i)31-s − 1.00i·36-s + 1.41·37-s − 1.00i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (668, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.661 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4410441409\)
\(L(\frac12)\) \(\approx\) \(0.4410441409\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
13 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1 - i)T - iT^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - 1.41iT - T^{2} \)
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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

−10.12333517881023246442380808469, −9.609935961658284939301281225758, −8.688531486729288316289045687590, −7.77883123159759318881216986027, −6.79568111740513100703927668871, −6.05393226072990700865975468828, −5.29979939129015066809749770709, −4.10804935203376750950095914495, −3.12912370675045720992234644539, −1.30013824168700798835663009302, 0.53989863197631150241351760767, 3.18158882225541922878341032721, 3.78103290608432380413934186731, 4.89906897595268344817712762185, 5.70658184755704858677148249742, 6.42412324622173685163668772045, 7.56906470391190801923500797307, 8.807859933682474104571846384335, 9.418883838324678918720880659088, 9.943231532288628334617655219877

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