Properties

Label 2-75-25.11-c3-0-11
Degree $2$
Conductor $75$
Sign $0.393 + 0.919i$
Analytic cond. $4.42514$
Root an. cond. $2.10360$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.67 − 2.67i)2-s + (−0.927 + 2.85i)3-s + (3.91 − 12.0i)4-s + (6.73 − 8.92i)5-s + (4.21 + 12.9i)6-s + 9.62·7-s + (−6.54 − 20.1i)8-s + (−7.28 − 5.29i)9-s + (0.908 − 50.8i)10-s + (−4.49 + 3.26i)11-s + (30.7 + 22.3i)12-s + (−39.0 − 28.3i)13-s + (35.4 − 25.7i)14-s + (19.2 + 27.4i)15-s + (4.03 + 2.92i)16-s + (29.0 + 89.3i)17-s + ⋯
L(s)  = 1  + (1.30 − 0.944i)2-s + (−0.178 + 0.549i)3-s + (0.489 − 1.50i)4-s + (0.602 − 0.798i)5-s + (0.286 + 0.882i)6-s + 0.519·7-s + (−0.289 − 0.890i)8-s + (−0.269 − 0.195i)9-s + (0.0287 − 1.60i)10-s + (−0.123 + 0.0895i)11-s + (0.739 + 0.537i)12-s + (−0.832 − 0.605i)13-s + (0.676 − 0.491i)14-s + (0.330 + 0.473i)15-s + (0.0629 + 0.0457i)16-s + (0.414 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.393 + 0.919i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.393 + 0.919i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.40695 - 1.58858i\)
\(L(\frac12)\) \(\approx\) \(2.40695 - 1.58858i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.927 - 2.85i)T \)
5 \( 1 + (-6.73 + 8.92i)T \)
good2 \( 1 + (-3.67 + 2.67i)T + (2.47 - 7.60i)T^{2} \)
7 \( 1 - 9.62T + 343T^{2} \)
11 \( 1 + (4.49 - 3.26i)T + (411. - 1.26e3i)T^{2} \)
13 \( 1 + (39.0 + 28.3i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-29.0 - 89.3i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (-14.5 - 44.6i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + (143. - 104. i)T + (3.75e3 - 1.15e4i)T^{2} \)
29 \( 1 + (20.1 - 61.8i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (-37.2 - 114. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (6.32 + 4.59i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (390. + 283. i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 421.T + 7.95e4T^{2} \)
47 \( 1 + (-134. + 414. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (-4.44 + 13.6i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (430. + 312. i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (-380. + 276. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + (-81.8 - 251. i)T + (-2.43e5 + 1.76e5i)T^{2} \)
71 \( 1 + (-60.2 + 185. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (66.1 - 48.0i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-379. + 1.16e3i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (-329. - 1.01e3i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + (941. - 683. i)T + (2.17e5 - 6.70e5i)T^{2} \)
97 \( 1 + (-351. + 1.08e3i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−13.75616904141857597395815445295, −12.56354148720396175425178798599, −12.01683828506411270625412159373, −10.61845861941733962320167488209, −9.866653134807516236052665495666, −8.200870016597521017331751293816, −5.77305852221869557085466064338, −5.03956444631272057157292637860, −3.74251143498605475735763857487, −1.83596395778117514518332950799, 2.63512541549245277615230536164, 4.63983527616140013043557878705, 5.86893901135965414084830727167, 6.89419792766851483960188217057, 7.77971418113166063670340978080, 9.772450562993552644820196370387, 11.39174260604409999850223554343, 12.33995047203763639836070763302, 13.63277787783074056644292262926, 14.13266348467045208221513595518

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