Properties

Label 2-675-135.4-c1-0-4
Degree $2$
Conductor $675$
Sign $-0.999 - 0.0330i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  + (−0.274 + 0.753i)2-s + (−0.386 + 1.68i)3-s + (1.03 + 0.872i)4-s + (−1.16 − 0.753i)6-s + (1.52 + 1.82i)7-s + (−2.33 + 1.34i)8-s + (−2.70 − 1.30i)9-s + (−0.0434 − 0.246i)11-s + (−1.87 + 1.41i)12-s + (0.893 + 2.45i)13-s + (−1.79 + 0.651i)14-s + (0.0969 + 0.549i)16-s + (0.254 + 0.146i)17-s + (1.72 − 1.67i)18-s + (−1.39 − 2.41i)19-s + ⋯
L(s)  = 1  + (−0.193 + 0.532i)2-s + (−0.223 + 0.974i)3-s + (0.519 + 0.436i)4-s + (−0.475 − 0.307i)6-s + (0.577 + 0.688i)7-s + (−0.823 + 0.475i)8-s + (−0.900 − 0.434i)9-s + (−0.0130 − 0.0742i)11-s + (−0.541 + 0.409i)12-s + (0.247 + 0.680i)13-s + (−0.478 + 0.174i)14-s + (0.0242 + 0.137i)16-s + (0.0616 + 0.0355i)17-s + (0.406 − 0.395i)18-s + (−0.319 − 0.553i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.999 - 0.0330i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.999 - 0.0330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0211137 + 1.27616i\)
\(L(\frac12)\) \(\approx\) \(0.0211137 + 1.27616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.386 - 1.68i)T \)
5 \( 1 \)
good2 \( 1 + (0.274 - 0.753i)T + (-1.53 - 1.28i)T^{2} \)
7 \( 1 + (-1.52 - 1.82i)T + (-1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.0434 + 0.246i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-0.893 - 2.45i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.254 - 0.146i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.39 + 2.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.30 - 5.12i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.333 + 0.121i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.11 - 1.77i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-6.05 - 3.49i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.13 + 3.32i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.256 + 0.0452i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (7.34 + 8.75i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + 5.43iT - 53T^{2} \)
59 \( 1 + (1.03 - 5.88i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (9.07 - 7.61i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.619 + 1.70i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-0.185 + 0.320i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.35 - 2.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.754 - 0.274i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.942 + 2.58i)T + (-63.5 - 53.3i)T^{2} \)
89 \( 1 + (-5.22 - 9.05i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.6 + 2.57i)T + (91.1 - 33.1i)T^{2} \)
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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

−11.08691050582023924521221129630, −9.972375887815244726643376177182, −9.011605849856507940268794317561, −8.466890884840038366497078456864, −7.50770622864547138223693288996, −6.33811688069957336179278183596, −5.66369552337431678395424859321, −4.58301215831128354386119070364, −3.44796471612703818254741366202, −2.22719716273728012237652568423, 0.72811502341204974006428134302, 1.84946039473102370694919236165, 2.95324164210154251672800103632, 4.48780885510793362106366246062, 5.89564862868922144945461552725, 6.37153842725196452232566539257, 7.59319889275039734973326139008, 8.059455156069577729989456066745, 9.339143519907306482152754551265, 10.42519577290814768652912210704

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