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Properties

Label 2-975-195.164-c1-0-15
Degree $2$
Conductor $975$
Sign $-0.0469 - 0.998i$
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  + 1.73i·3-s − 2i·4-s + (3.36 + 3.36i)7-s − 2.99·9-s + 3.46·12-s + (−2.5 + 2.59i)13-s − 4·16-s + (3.83 + 3.83i)19-s + (−5.83 + 5.83i)21-s − 5.19i·27-s + (6.73 − 6.73i)28-s + (0.830 + 0.830i)31-s + 5.99i·36-s + (8.46 + 8.46i)37-s + (−4.5 − 4.33i)39-s + ⋯
L(s)  = 1  + 0.999i·3-s − i·4-s + (1.27 + 1.27i)7-s − 0.999·9-s + 1.00·12-s + (−0.693 + 0.720i)13-s − 16-s + (0.878 + 0.878i)19-s + (−1.27 + 1.27i)21-s − 0.999i·27-s + (1.27 − 1.27i)28-s + (0.149 + 0.149i)31-s + 0.999i·36-s + (1.39 + 1.39i)37-s + (−0.720 − 0.693i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0469 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0469 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.0469 - 0.998i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.0469 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07343 + 1.12509i\)
\(L(\frac12)\) \(\approx\) \(1.07343 + 1.12509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
5 \( 1 \)
13 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 + 2iT^{2} \)
7 \( 1 + (-3.36 - 3.36i)T + 7iT^{2} \)
11 \( 1 + 11iT^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + (-3.83 - 3.83i)T + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-0.830 - 0.830i)T + 31iT^{2} \)
37 \( 1 + (-8.46 - 8.46i)T + 37iT^{2} \)
41 \( 1 - 41iT^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 - 8.66T + 61T^{2} \)
67 \( 1 + (-5.29 + 5.29i)T - 67iT^{2} \)
71 \( 1 - 71iT^{2} \)
73 \( 1 + (-11.9 - 11.9i)T + 73iT^{2} \)
79 \( 1 + 17.3T + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (7.02 - 7.02i)T - 97iT^{2} \)
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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

−9.931761682012415751339718400394, −9.654459677126068125577758901936, −8.672352787210983952467260163745, −8.035052858320661709260629747371, −6.58255775500668703481249970398, −5.54417976903032211066140847410, −5.11544503807145638139873431174, −4.31469501421071280506919229548, −2.74064560888535312860770154905, −1.64940822734384121253973934642, 0.74105292342560875808386332519, 2.18812974138408261539050383892, 3.30439494823256701558797315501, 4.47434675028010134372696776975, 5.35891493113113922688340785406, 6.82197504579202096549246691743, 7.41523422672832380092571142305, 7.897459083002756728474596408339, 8.577231462851040107743406425138, 9.759250900689966068994941742042

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