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Properties

Label 2-975-65.32-c1-0-37
Degree $2$
Conductor $975$
Sign $-0.931 + 0.364i$
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  + (0.965 − 1.67i)2-s + (−0.258 − 0.965i)3-s + (−0.866 − 1.50i)4-s + (−1.86 − 0.500i)6-s + 0.517·8-s + (−0.866 + 0.499i)9-s + (5.96 − 1.59i)11-s + (−1.22 + 1.22i)12-s + (−3.60 + 0.0693i)13-s + (2.23 − 3.86i)16-s + (−5.98 − 1.60i)17-s + 1.93i·18-s + (1.90 − 7.09i)19-s + (3.08 − 11.5i)22-s + (−0.258 + 0.0693i)23-s + (−0.133 − 0.499i)24-s + ⋯
L(s)  = 1  + (0.683 − 1.18i)2-s + (−0.149 − 0.557i)3-s + (−0.433 − 0.750i)4-s + (−0.761 − 0.204i)6-s + 0.183·8-s + (−0.288 + 0.166i)9-s + (1.79 − 0.481i)11-s + (−0.353 + 0.353i)12-s + (−0.999 + 0.0192i)13-s + (0.558 − 0.966i)16-s + (−1.45 − 0.388i)17-s + 0.455i·18-s + (0.436 − 1.62i)19-s + (0.658 − 2.45i)22-s + (−0.0539 + 0.0144i)23-s + (−0.0273 − 0.102i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.931 + 0.364i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (682, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.931 + 0.364i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409692 - 2.17076i\)
\(L(\frac12)\) \(\approx\) \(0.409692 - 2.17076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
13 \( 1 + (3.60 - 0.0693i)T \)
good2 \( 1 + (-0.965 + 1.67i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.96 + 1.59i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (5.98 + 1.60i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.90 + 7.09i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.258 - 0.0693i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.56 + 2.63i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 + 3i)T - 31iT^{2} \)
37 \( 1 + (-5.91 - 3.41i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.46 - 9.19i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.0507 - 0.189i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 1.41iT - 47T^{2} \)
53 \( 1 + (0.378 - 0.378i)T - 53iT^{2} \)
59 \( 1 + (-0.633 - 0.169i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.59 + 7.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.22 - 2.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.3 - 3.03i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 3.34T + 73T^{2} \)
79 \( 1 - 5.26iT - 79T^{2} \)
83 \( 1 + 17.1iT - 83T^{2} \)
89 \( 1 + (-2.16 - 8.09i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.31 - 7.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.556021840630014496704144291836, −9.243539800997825047186387031309, −7.901749349593165604204849992207, −6.93255697521440866958238970265, −6.24424782715985958318293662132, −4.84676781224506887694925549612, −4.26774108946031776568130701787, −3.01740509548050988017856926591, −2.16548757737880142270795854640, −0.874093627769907127937867656987, 1.83371658107266417907930956306, 3.76281720891288301244173765406, 4.28102759595718499596886398601, 5.22708575597890071102358680964, 6.12931446621593702875426105318, 6.80247487759674467957935352886, 7.56781587711021171345331542658, 8.651162023947260557438218662756, 9.434750901769840351996731646634, 10.25413972909589758789390923558

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