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Properties

Label 2-975-13.3-c1-0-37
Degree $2$
Conductor $975$
Sign $-0.0128 + 0.999i$
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  + (0.5 + 0.866i)3-s + (1 − 1.73i)4-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−3 − 5.19i)11-s + 1.99·12-s + (−2.5 − 2.59i)13-s + (−1.99 − 3.46i)16-s + (2 − 3.46i)19-s − 0.999·21-s + (−3 − 5.19i)23-s − 0.999·27-s + (0.999 + 1.73i)28-s + (3 + 5.19i)29-s + 5·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.5 − 0.866i)4-s + (−0.188 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.904 − 1.56i)11-s + 0.577·12-s + (−0.693 − 0.720i)13-s + (−0.499 − 0.866i)16-s + (0.458 − 0.794i)19-s − 0.218·21-s + (−0.625 − 1.08i)23-s − 0.192·27-s + (0.188 + 0.327i)28-s + (0.557 + 0.964i)29-s + 0.898·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.0128 + 0.999i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01920 - 1.03236i\)
\(L(\frac12)\) \(\approx\) \(1.01920 - 1.03236i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5T + 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.02975022624412529658145835510, −8.966804010693216686448943160908, −8.293223098769118974460548232596, −7.26589658632813149539282888599, −6.16616187220828141975431569214, −5.48373856641023547758544326878, −4.70827077902163495468691649037, −3.09111252477138991676060885631, −2.52422111222371175233975026980, −0.60545402925595059545070026678, 1.86222805469079783834687105536, 2.67624857824139474609539248749, 3.87816146274206125123303765884, 4.81807256052685982540541260740, 6.20966785527238263166266526435, 7.08718959897244533879968835452, 7.67728822500440549572714270506, 8.172564121722741717254330400718, 9.590988711530504816972664799212, 9.956536734480832762278934354001

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