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Properties

Label 2-975-13.10-c1-0-12
Degree $2$
Conductor $975$
Sign $0.252 - 0.967i$
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 + (3 + 1.73i)7-s + (−0.499 + 0.866i)9-s + 1.99·12-s + (2.5 − 2.59i)13-s + (−1.99 − 3.46i)16-s + (−3 + 5.19i)17-s + (−1.5 − 0.866i)19-s − 3.46i·21-s + (3 + 5.19i)23-s + 0.999·27-s + (−6 + 3.46i)28-s + (3 + 5.19i)29-s − 5.19i·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (1.13 + 0.654i)7-s + (−0.166 + 0.288i)9-s + 0.577·12-s + (0.693 − 0.720i)13-s + (−0.499 − 0.866i)16-s + (−0.727 + 1.26i)17-s + (−0.344 − 0.198i)19-s − 0.755i·21-s + (0.625 + 1.08i)23-s + 0.192·27-s + (−1.13 + 0.654i)28-s + (0.557 + 0.964i)29-s − 0.933i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00672 + 0.777628i\)
\(L(\frac12)\) \(\approx\) \(1.00672 + 0.777628i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 - 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
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 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (9 - 5.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-9 - 5.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.66iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (9 - 5.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.5 - 7.79i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.29273330238597684440856561400, −8.970581611993185185369768182012, −8.424062741516042919181367775091, −7.900688629047452114625538474152, −6.89473734970337166177236000032, −5.80116215371448371432923662211, −4.97524432430367373473595144165, −3.97920720334513990865078097038, −2.76547040015412949209490289863, −1.45213911259607608662791721562, 0.67518652001974141877248812394, 2.03804722544250792827156278307, 3.86283752067353597273966652860, 4.70129807143628140411995943723, 5.16543727756025293640066872188, 6.40255826047643545743538754590, 7.12129245639779337268539584476, 8.620876657791883854407544653642, 8.810397854892319574645981941990, 10.09683535144635687611820333877

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