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Properties

Label 2-875-35.13-c1-0-60
Degree $2$
Conductor $875$
Sign $-0.895 + 0.444i$
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  + (1.67 − 1.67i)2-s + (2.17 − 2.17i)3-s − 3.61i·4-s − 7.29i·6-s + (−0.844 + 2.50i)7-s + (−2.71 − 2.71i)8-s − 6.47i·9-s + 2.61·11-s + (−7.87 − 7.87i)12-s + (−1.02 + 1.02i)13-s + (2.78 + 5.61i)14-s − 1.85·16-s + (−0.831 − 0.831i)17-s + (−10.8 − 10.8i)18-s − 7.29·19-s + ⋯
L(s)  = 1  + (1.18 − 1.18i)2-s + (1.25 − 1.25i)3-s − 1.80i·4-s − 2.97i·6-s + (−0.319 + 0.947i)7-s + (−0.958 − 0.958i)8-s − 2.15i·9-s + 0.789·11-s + (−2.27 − 2.27i)12-s + (−0.284 + 0.284i)13-s + (0.744 + 1.50i)14-s − 0.463·16-s + (−0.201 − 0.201i)17-s + (−2.55 − 2.55i)18-s − 1.67·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(6.98691\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (818, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.958729 - 4.09076i\)
\(L(\frac12)\) \(\approx\) \(0.958729 - 4.09076i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.844 - 2.50i)T \)
good2 \( 1 + (-1.67 + 1.67i)T - 2iT^{2} \)
3 \( 1 + (-2.17 + 2.17i)T - 3iT^{2} \)
11 \( 1 - 2.61T + 11T^{2} \)
13 \( 1 + (1.02 - 1.02i)T - 13iT^{2} \)
17 \( 1 + (0.831 + 0.831i)T + 17iT^{2} \)
19 \( 1 + 7.29T + 19T^{2} \)
23 \( 1 + (-5.02 - 5.02i)T + 23iT^{2} \)
29 \( 1 - 8.70iT - 29T^{2} \)
31 \( 1 + 1.06iT - 31T^{2} \)
37 \( 1 + (-3.10 + 3.10i)T - 37iT^{2} \)
41 \( 1 - 5.57iT - 41T^{2} \)
43 \( 1 + (1.03 + 1.03i)T + 43iT^{2} \)
47 \( 1 + (5.37 + 5.37i)T + 47iT^{2} \)
53 \( 1 + (3.99 + 3.99i)T + 53iT^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 - 7.95iT - 61T^{2} \)
67 \( 1 + (0.640 - 0.640i)T - 67iT^{2} \)
71 \( 1 - 1.76T + 71T^{2} \)
73 \( 1 + (-10.4 + 10.4i)T - 73iT^{2} \)
79 \( 1 - 3.70iT - 79T^{2} \)
83 \( 1 + (3.20 - 3.20i)T - 83iT^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (1.02 + 1.02i)T + 97iT^{2} \)
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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

−9.677175259005608896912480417782, −9.025637321204586395884747861856, −8.293139778689847808515203973605, −6.98199894959915431528207626797, −6.35733402621809587154458172014, −5.18452870968399870100942729568, −3.90235804454047618171699198961, −3.06635428629353774344432295766, −2.26308287986561004967232111913, −1.46470011930566255211256503101, 2.62945049483515324493667451688, 3.69970405218588705564026717710, 4.26794455838272757492399576628, 4.81997104060353376638885447718, 6.22380859783111544405267925514, 6.94861824463455409509329372811, 8.009162220615031051955271623001, 8.543857996676852979686574810408, 9.565331256053146876626101509009, 10.32439579521352119677203872858

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