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Properties

Label 2-975-65.29-c1-0-7
Degree $2$
Conductor $975$
Sign $-0.187 - 0.982i$
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  + (−1.35 + 0.780i)2-s + (−0.866 + 0.5i)3-s + (0.219 − 0.379i)4-s + (0.780 − 1.35i)6-s + (−0.486 − 0.280i)7-s − 2.43i·8-s + (0.499 − 0.866i)9-s + (1 + 1.73i)11-s + 0.438i·12-s + (−3.57 − 0.5i)13-s + 0.876·14-s + (2.34 + 4.05i)16-s + (1.35 + 0.780i)17-s + 1.56i·18-s + (3.56 − 6.16i)19-s + ⋯
L(s)  = 1  + (−0.956 + 0.552i)2-s + (−0.499 + 0.288i)3-s + (0.109 − 0.189i)4-s + (0.318 − 0.552i)6-s + (−0.183 − 0.106i)7-s − 0.862i·8-s + (0.166 − 0.288i)9-s + (0.301 + 0.522i)11-s + 0.126i·12-s + (−0.990 − 0.138i)13-s + 0.234·14-s + (0.585 + 1.01i)16-s + (0.327 + 0.189i)17-s + 0.368i·18-s + (0.817 − 1.41i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.187 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396553 + 0.479616i\)
\(L(\frac12)\) \(\approx\) \(0.396553 + 0.479616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (3.57 + 0.5i)T \)
good2 \( 1 + (1.35 - 0.780i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (0.486 + 0.280i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.35 - 0.780i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.56 + 6.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.73 - i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.34 - 5.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.56T + 31T^{2} \)
37 \( 1 + (-6.54 + 3.78i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.780 - 1.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.95 - 2.28i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.24iT - 47T^{2} \)
53 \( 1 - 0.684iT - 53T^{2} \)
59 \( 1 + (1.43 - 2.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.93 - 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.95 + 2.28i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7 - 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.1iT - 73T^{2} \)
79 \( 1 + 5.43T + 79T^{2} \)
83 \( 1 - 0.876iT - 83T^{2} \)
89 \( 1 + (-2.43 - 4.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.41 - 4.28i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.854813674831105063481695290597, −9.542926233327313899725606813778, −8.667749393986982771996650190658, −7.58068064232644239043429161992, −7.10062708856696410792384222242, −6.22079178725782736450490556344, −5.05786965159962229997870517421, −4.17441460762045066856191214947, −2.86426513479595667214727981572, −0.951381629993696526030430067783, 0.57979410060995197538932654271, 1.83913219480542443789990518579, 3.02920919942365644238000170375, 4.52131322438891955946783497185, 5.57969419306722613903334213356, 6.29488531320391147957558998020, 7.58375715778615504290467321715, 8.105393724804534865423443388207, 9.158068010672461769470106386988, 9.932782187307198008140292835266

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