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Properties

Label 2-975-5.4-c1-0-0
Degree $2$
Conductor $975$
Sign $0.447 - 0.894i$
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  − 2.48i·2-s + i·3-s − 4.19·4-s + 2.48·6-s − 1.19i·7-s + 5.46i·8-s − 9-s − 1.19·11-s − 4.19i·12-s − i·13-s − 2.97·14-s + 5.21·16-s + 6.17i·17-s + 2.48i·18-s − 6.97·19-s + ⋯
L(s)  = 1  − 1.76i·2-s + 0.577i·3-s − 2.09·4-s + 1.01·6-s − 0.452i·7-s + 1.93i·8-s − 0.333·9-s − 0.360·11-s − 1.21i·12-s − 0.277i·13-s − 0.796·14-s + 1.30·16-s + 1.49i·17-s + 0.586i·18-s − 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111731 + 0.0690539i\)
\(L(\frac12)\) \(\approx\) \(0.111731 + 0.0690539i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
13 \( 1 + iT \)
good2 \( 1 + 2.48iT - 2T^{2} \)
7 \( 1 + 1.19iT - 7T^{2} \)
11 \( 1 + 1.19T + 11T^{2} \)
17 \( 1 - 6.17iT - 17T^{2} \)
19 \( 1 + 6.97T + 19T^{2} \)
23 \( 1 + 4.17iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 2.97T + 31T^{2} \)
37 \( 1 - 7.78iT - 37T^{2} \)
41 \( 1 + 6.17T + 41T^{2} \)
43 \( 1 - 9.95iT - 43T^{2} \)
47 \( 1 + 1.02iT - 47T^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 + 5.37T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 - 9.37iT - 67T^{2} \)
71 \( 1 + 5.19T + 71T^{2} \)
73 \( 1 - 11.9iT - 73T^{2} \)
79 \( 1 - 1.78T + 79T^{2} \)
83 \( 1 - 5.37iT - 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + 1.82iT - 97T^{2} \)
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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

−10.23747172699111496466814559363, −9.841532883010060604669376265918, −8.616077350901613521496146708988, −8.269399661457781007960468908097, −6.65739091274957298006745735864, −5.44908867718064415994054435779, −4.33745900979908265966801562894, −3.85096843753709715796561504672, −2.74589052912346347466346002477, −1.64750551038587228950260744757, 0.05889724693817678428696016877, 2.24072169709271762784196079828, 3.88456014434641877908589326214, 5.08393976262261890803366787068, 5.67680850964086649547778111379, 6.57565638892244278597245708981, 7.31948693479767534053386853108, 7.85004914491059140278042041109, 9.028551265913521836087276876695, 9.122731980486811315230740012853

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