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Properties

Label 2-975-195.164-c1-0-36
Degree $2$
Conductor $975$
Sign $-0.994 - 0.106i$
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  + (−1.11 − 1.11i)2-s + (−1.67 − 0.455i)3-s + 0.503i·4-s + (1.36 + 2.37i)6-s + (1.46 + 1.46i)7-s + (−1.67 + 1.67i)8-s + (2.58 + 1.52i)9-s + (0.292 + 0.292i)11-s + (0.229 − 0.841i)12-s + (−2.86 − 2.19i)13-s − 3.28i·14-s + 4.75·16-s + 2.78i·17-s + (−1.18 − 4.59i)18-s + (1.21 + 1.21i)19-s + ⋯
L(s)  = 1  + (−0.791 − 0.791i)2-s + (−0.964 − 0.262i)3-s + 0.251i·4-s + (0.555 + 0.971i)6-s + (0.554 + 0.554i)7-s + (−0.591 + 0.591i)8-s + (0.861 + 0.507i)9-s + (0.0883 + 0.0883i)11-s + (0.0661 − 0.242i)12-s + (−0.793 − 0.608i)13-s − 0.876i·14-s + 1.18·16-s + 0.674i·17-s + (−0.280 − 1.08i)18-s + (0.277 + 0.277i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.994 - 0.106i$
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: \((2,\ 975,\ (\ :1/2),\ -0.994 - 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0194908 + 0.365596i\)
\(L(\frac12)\) \(\approx\) \(0.0194908 + 0.365596i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 + 0.455i)T \)
5 \( 1 \)
13 \( 1 + (2.86 + 2.19i)T \)
good2 \( 1 + (1.11 + 1.11i)T + 2iT^{2} \)
7 \( 1 + (-1.46 - 1.46i)T + 7iT^{2} \)
11 \( 1 + (-0.292 - 0.292i)T + 11iT^{2} \)
17 \( 1 - 2.78iT - 17T^{2} \)
19 \( 1 + (-1.21 - 1.21i)T + 19iT^{2} \)
23 \( 1 + 5.66iT - 23T^{2} \)
29 \( 1 + 7.34iT - 29T^{2} \)
31 \( 1 + (1.98 + 1.98i)T + 31iT^{2} \)
37 \( 1 + (3.02 + 3.02i)T + 37iT^{2} \)
41 \( 1 + (3.08 - 3.08i)T - 41iT^{2} \)
43 \( 1 + 0.831T + 43T^{2} \)
47 \( 1 + (-8.76 + 8.76i)T - 47iT^{2} \)
53 \( 1 - 0.258T + 53T^{2} \)
59 \( 1 + (2.38 + 2.38i)T + 59iT^{2} \)
61 \( 1 + 3.66T + 61T^{2} \)
67 \( 1 + (9.29 - 9.29i)T - 67iT^{2} \)
71 \( 1 + (7.88 - 7.88i)T - 71iT^{2} \)
73 \( 1 + (-11.5 - 11.5i)T + 73iT^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 + (4.97 + 4.97i)T + 89iT^{2} \)
97 \( 1 + (-8.41 + 8.41i)T - 97iT^{2} \)
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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

−10.05722795457367837273737775072, −8.804848793817848096362648493242, −8.111186826371271605302606970419, −7.13942518643777661479054752788, −5.92384890990244465834219124728, −5.41607831167397274659192354250, −4.29036003654647384876118886131, −2.58783570725969919356141581751, −1.66672148569207233065450869934, −0.27569558696278562979420392570, 1.28834683433758671845296394303, 3.34543155254039782976365576134, 4.54139437184045079062764119691, 5.34853709875050949600547577034, 6.40016624957763028182102690165, 7.31917166108987974686699763162, 7.50647052491876392233862324069, 8.943690696265786496972721069591, 9.420697749291324788345314185824, 10.33352787819101245527108080727

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