Properties

Label 2-75-25.6-c1-0-5
Degree $2$
Conductor $75$
Sign $-0.915 + 0.403i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
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  + (−0.830 − 2.55i)2-s + (0.809 − 0.587i)3-s + (−4.22 + 3.06i)4-s + (−1.34 − 1.78i)5-s + (−2.17 − 1.57i)6-s + 1.68·7-s + (7.00 + 5.08i)8-s + (0.309 − 0.951i)9-s + (−3.43 + 4.92i)10-s + (0.333 + 1.02i)11-s + (−1.61 + 4.96i)12-s + (0.827 − 2.54i)13-s + (−1.40 − 4.31i)14-s + (−2.13 − 0.650i)15-s + (3.95 − 12.1i)16-s + (3.18 + 2.31i)17-s + ⋯
L(s)  = 1  + (−0.587 − 1.80i)2-s + (0.467 − 0.339i)3-s + (−2.11 + 1.53i)4-s + (−0.603 − 0.797i)5-s + (−0.887 − 0.644i)6-s + 0.637·7-s + (2.47 + 1.79i)8-s + (0.103 − 0.317i)9-s + (−1.08 + 1.55i)10-s + (0.100 + 0.309i)11-s + (−0.465 + 1.43i)12-s + (0.229 − 0.706i)13-s + (−0.374 − 1.15i)14-s + (−0.552 − 0.167i)15-s + (0.989 − 3.04i)16-s + (0.771 + 0.560i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.915 + 0.403i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ -0.915 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146828 - 0.697504i\)
\(L(\frac12)\) \(\approx\) \(0.146828 - 0.697504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (1.34 + 1.78i)T \)
good2 \( 1 + (0.830 + 2.55i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 - 1.68T + 7T^{2} \)
11 \( 1 + (-0.333 - 1.02i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.827 + 2.54i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.18 - 2.31i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.952 - 0.692i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.25 + 3.86i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.81 - 3.50i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-5.74 - 4.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.41 - 4.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.47 - 10.7i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.58T + 43T^{2} \)
47 \( 1 + (-1.55 + 1.12i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.06 + 1.49i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.412 + 1.27i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.24 + 6.92i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (10.0 + 7.31i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-4.84 + 3.51i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.01 + 3.13i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.23 + 2.35i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.17 + 5.21i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.77 - 14.6i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.71 - 6.32i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.55250421159178264073043734147, −12.57107920537548141958803766911, −11.99229762371828103969308281934, −10.83335269520862955904549217705, −9.684425215816149560305929849683, −8.459096484498833314182327473491, −7.923972461286391671602772152923, −4.75085177306162577313623775268, −3.35088512155409181807641099331, −1.41239068000071307683846019345, 4.12523116994402566546215881704, 5.69120716025544440932950658308, 7.16043764029155767114305469739, 7.894122445919307159623635702474, 8.998427412928074920497902181079, 10.10727313033129376302053926359, 11.50018599629825737779475459242, 13.75038908246752268344421256278, 14.27444440464199051398478104985, 15.20937971804439282111021104544

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