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Properties

Label 2-875-175.17-c1-0-17
Degree $2$
Conductor $875$
Sign $-0.471 - 0.882i$
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  + (1.31 − 0.0691i)2-s + (−0.985 + 1.21i)3-s + (−0.253 + 0.0266i)4-s + (−1.21 + 1.67i)6-s + (2.58 − 0.569i)7-s + (−2.94 + 0.465i)8-s + (0.113 + 0.536i)9-s + (0.468 + 0.0995i)11-s + (0.217 − 0.335i)12-s + (−0.195 + 0.0993i)13-s + (3.36 − 0.930i)14-s + (−3.34 + 0.712i)16-s + (−1.72 + 4.48i)17-s + (0.187 + 0.699i)18-s + (−0.561 + 5.33i)19-s + ⋯
L(s)  = 1  + (0.932 − 0.0488i)2-s + (−0.568 + 0.702i)3-s + (−0.126 + 0.0133i)4-s + (−0.496 + 0.683i)6-s + (0.976 − 0.215i)7-s + (−1.04 + 0.164i)8-s + (0.0379 + 0.178i)9-s + (0.141 + 0.0300i)11-s + (0.0628 − 0.0967i)12-s + (−0.0540 + 0.0275i)13-s + (0.900 − 0.248i)14-s + (−0.837 + 0.178i)16-s + (−0.417 + 1.08i)17-s + (0.0441 + 0.164i)18-s + (−0.128 + 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $-0.471 - 0.882i$
Analytic conductor: \(6.98691\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :1/2),\ -0.471 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.784217 + 1.30789i\)
\(L(\frac12)\) \(\approx\) \(0.784217 + 1.30789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.58 + 0.569i)T \)
good2 \( 1 + (-1.31 + 0.0691i)T + (1.98 - 0.209i)T^{2} \)
3 \( 1 + (0.985 - 1.21i)T + (-0.623 - 2.93i)T^{2} \)
11 \( 1 + (-0.468 - 0.0995i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (0.195 - 0.0993i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (1.72 - 4.48i)T + (-12.6 - 11.3i)T^{2} \)
19 \( 1 + (0.561 - 5.33i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (-0.0278 - 0.532i)T + (-22.8 + 2.40i)T^{2} \)
29 \( 1 + (-2.22 - 3.06i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.36 - 3.05i)T + (-20.7 + 23.0i)T^{2} \)
37 \( 1 + (6.39 + 4.15i)T + (15.0 + 33.8i)T^{2} \)
41 \( 1 + (2.67 + 0.870i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (6.03 - 6.03i)T - 43iT^{2} \)
47 \( 1 + (6.98 - 2.68i)T + (34.9 - 31.4i)T^{2} \)
53 \( 1 + (3.76 + 3.04i)T + (11.0 + 51.8i)T^{2} \)
59 \( 1 + (-3.58 - 3.97i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-6.61 - 5.95i)T + (6.37 + 60.6i)T^{2} \)
67 \( 1 + (-12.1 - 4.65i)T + (49.7 + 44.8i)T^{2} \)
71 \( 1 + (-3.80 + 2.76i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-7.14 - 11.0i)T + (-29.6 + 66.6i)T^{2} \)
79 \( 1 + (-3.76 + 8.46i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (1.48 + 9.38i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-3.91 + 4.35i)T + (-9.30 - 88.5i)T^{2} \)
97 \( 1 + (1.50 - 9.51i)T + (-92.2 - 29.9i)T^{2} \)
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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

−10.53648761037209411000956006898, −9.846550432973491805606428757088, −8.622573430463747830734023422370, −8.061246870767386286784271537102, −6.68499124624763563002228207828, −5.66310907235190971111449996543, −5.02426505612079096894023912568, −4.28820638584397487993754669706, −3.53152513109967780085728318574, −1.83163411461813997424428498091, 0.58938630584483001777737322767, 2.25139718510748579697255341540, 3.59181240090333671877857433021, 4.84727479823566908495308254679, 5.18742548913469764801132301332, 6.41752173420437437936303397591, 6.91006294992871733077314289205, 8.126544017105066224134290871370, 8.982809195127308997662975611436, 9.815178648431229424898365158323

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