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Properties

Label 2-875-25.6-c1-0-15
Degree $2$
Conductor $875$
Sign $-0.893 - 0.449i$
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  + (0.686 + 2.11i)2-s + (0.708 − 0.515i)3-s + (−2.37 + 1.72i)4-s + (1.57 + 1.14i)6-s + 7-s + (−1.69 − 1.22i)8-s + (−0.689 + 2.12i)9-s + (−0.121 − 0.373i)11-s + (−0.796 + 2.45i)12-s + (−0.715 + 2.20i)13-s + (0.686 + 2.11i)14-s + (−0.380 + 1.17i)16-s + (−2.07 − 1.50i)17-s − 4.96·18-s + (2.59 + 1.88i)19-s + ⋯
L(s)  = 1  + (0.485 + 1.49i)2-s + (0.409 − 0.297i)3-s + (−1.18 + 0.864i)4-s + (0.643 + 0.467i)6-s + 0.377·7-s + (−0.598 − 0.434i)8-s + (−0.229 + 0.707i)9-s + (−0.0366 − 0.112i)11-s + (−0.229 + 0.707i)12-s + (−0.198 + 0.610i)13-s + (0.183 + 0.564i)14-s + (−0.0952 + 0.293i)16-s + (−0.503 − 0.366i)17-s − 1.16·18-s + (0.594 + 0.431i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $-0.893 - 0.449i$
Analytic conductor: \(6.98691\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :1/2),\ -0.893 - 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.496349 + 2.08952i\)
\(L(\frac12)\) \(\approx\) \(0.496349 + 2.08952i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + (-0.686 - 2.11i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.708 + 0.515i)T + (0.927 - 2.85i)T^{2} \)
11 \( 1 + (0.121 + 0.373i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.715 - 2.20i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.07 + 1.50i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.59 - 1.88i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.921 - 2.83i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.59 + 1.15i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-7.28 - 5.28i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.86 - 5.74i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.76 + 5.44i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.34T + 43T^{2} \)
47 \( 1 + (2.47 - 1.79i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-8.34 + 6.06i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.53 + 4.71i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.30 + 10.1i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (12.7 + 9.25i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-9.95 + 7.23i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.413 + 1.27i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.855 - 0.621i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-11.5 - 8.36i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.34 - 13.3i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-13.7 + 10.0i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.43397764263849389886549076924, −9.273485181348413179613657105809, −8.380898905053950350722211297485, −7.902504487720243731303579396165, −7.04012964780423808102342137292, −6.36856733530417100201837387620, −5.16059835107456926649454778618, −4.75837137417878712293584883947, −3.36908777007243907673146314921, −1.90043387647320425373315719292, 0.879102697997028612677612731309, 2.38034558820707636232821134518, 3.14228912004500821233959398292, 4.13874087713146738833580640352, 4.88431208690866615875536643604, 6.06376523736919896783835408902, 7.30854745074432203982204463307, 8.474876095086160523306240786024, 9.185313327229383601264253248067, 10.09649209077863323246059942810

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