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Properties

Label 2-875-25.4-c1-0-36
Degree $2$
Conductor $875$
Sign $-0.988 + 0.153i$
Analytic cond. $6.98691$
Root an. cond. $2.64327$
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  + (−1.41 − 0.459i)2-s + (1.82 − 2.50i)3-s + (0.167 + 0.121i)4-s + (−3.72 + 2.70i)6-s − i·7-s + (1.56 + 2.15i)8-s + (−2.04 − 6.29i)9-s + (1.87 − 5.76i)11-s + (0.611 − 0.198i)12-s + (3.03 − 0.984i)13-s + (−0.459 + 1.41i)14-s + (−1.35 − 4.15i)16-s + (1.80 + 2.47i)17-s + 9.83i·18-s + (−1.24 + 0.901i)19-s + ⋯
L(s)  = 1  + (−0.999 − 0.324i)2-s + (1.05 − 1.44i)3-s + (0.0837 + 0.0608i)4-s + (−1.52 + 1.10i)6-s − 0.377i·7-s + (0.553 + 0.761i)8-s + (−0.682 − 2.09i)9-s + (0.565 − 1.73i)11-s + (0.176 − 0.0573i)12-s + (0.840 − 0.273i)13-s + (−0.122 + 0.377i)14-s + (−0.337 − 1.03i)16-s + (0.436 + 0.601i)17-s + 2.31i·18-s + (−0.284 + 0.206i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $-0.988 + 0.153i$
Analytic conductor: \(6.98691\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :1/2),\ -0.988 + 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0957556 - 1.24055i\)
\(L(\frac12)\) \(\approx\) \(0.0957556 - 1.24055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + iT \)
good2 \( 1 + (1.41 + 0.459i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.82 + 2.50i)T + (-0.927 - 2.85i)T^{2} \)
11 \( 1 + (-1.87 + 5.76i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-3.03 + 0.984i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.80 - 2.47i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.24 - 0.901i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.430 + 0.139i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-2.38 - 1.73i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.913 - 0.663i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.90 + 1.26i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.45 + 7.54i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.03iT - 43T^{2} \)
47 \( 1 + (3.40 - 4.68i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.36 - 3.25i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.23 + 6.86i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.33 - 13.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-5.76 - 7.93i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.06 + 3.68i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.41 + 0.784i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.78 - 2.02i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.5 - 14.4i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.615 + 1.89i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.46 - 4.77i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.413145716965400905537603037049, −8.644182963966665177225155255886, −8.305726873394462100814224915800, −7.60758605973759389738291901820, −6.51054001122629694168526448903, −5.75664445129912439816648560370, −3.83044195403774479000455480987, −2.89446020260068366186774041741, −1.53666263982876876644870794253, −0.820842000907214940370959909230, 1.90684229440423036520468440028, 3.30859830753139099736170983256, 4.28420419893120293164523461865, 4.87178777142303512685019939280, 6.50310865966893042328797521494, 7.56025453822504430781051573676, 8.302273485953270125803538298310, 9.020814155465461547444850595229, 9.630708061847173613199006150058, 9.964548085605104347282204941055

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