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Properties

Label 2-675-135.2-c1-0-34
Degree $2$
Conductor $675$
Sign $0.970 - 0.239i$
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  + (0.856 + 1.22i)2-s + (0.960 − 1.44i)3-s + (−0.0790 + 0.217i)4-s + (2.58 − 0.0591i)6-s + (4.36 + 2.03i)7-s + (2.55 − 0.683i)8-s + (−1.15 − 2.76i)9-s + (−2.25 + 2.68i)11-s + (0.237 + 0.322i)12-s + (−3.11 − 2.18i)13-s + (1.24 + 7.08i)14-s + (3.37 + 2.83i)16-s + (1.37 + 0.367i)17-s + (2.39 − 3.78i)18-s + (−1.30 − 0.750i)19-s + ⋯
L(s)  = 1  + (0.605 + 0.865i)2-s + (0.554 − 0.832i)3-s + (−0.0395 + 0.108i)4-s + (1.05 − 0.0241i)6-s + (1.65 + 0.769i)7-s + (0.902 − 0.241i)8-s + (−0.384 − 0.923i)9-s + (−0.678 + 0.808i)11-s + (0.0684 + 0.0931i)12-s + (−0.864 − 0.605i)13-s + (0.333 + 1.89i)14-s + (0.844 + 0.708i)16-s + (0.332 + 0.0891i)17-s + (0.565 − 0.892i)18-s + (−0.298 − 0.172i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.970 - 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86786 + 0.349136i\)
\(L(\frac12)\) \(\approx\) \(2.86786 + 0.349136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.960 + 1.44i)T \)
5 \( 1 \)
good2 \( 1 + (-0.856 - 1.22i)T + (-0.684 + 1.87i)T^{2} \)
7 \( 1 + (-4.36 - 2.03i)T + (4.49 + 5.36i)T^{2} \)
11 \( 1 + (2.25 - 2.68i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (3.11 + 2.18i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (-1.37 - 0.367i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.30 + 0.750i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.633 - 1.35i)T + (-14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.168 + 0.957i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.44 + 0.891i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-1.79 + 6.69i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.670 - 0.118i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.200 + 0.0175i)T + (42.3 - 7.46i)T^{2} \)
47 \( 1 + (3.68 - 7.89i)T + (-30.2 - 36.0i)T^{2} \)
53 \( 1 + (-2.81 - 2.81i)T + 53iT^{2} \)
59 \( 1 + (5.69 - 4.77i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.08 - 0.396i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-8.79 + 12.5i)T + (-22.9 - 62.9i)T^{2} \)
71 \( 1 + (11.1 - 6.42i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.0920 + 0.343i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.66 + 1.70i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (9.30 - 6.51i)T + (28.3 - 77.9i)T^{2} \)
89 \( 1 + (2.09 - 3.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.561 + 6.41i)T + (-95.5 + 16.8i)T^{2} \)
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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

−10.60457465021951881528352933499, −9.482850791419633419807563227928, −8.358072925114392848667095354222, −7.65548959076416025619060359684, −7.27151254665243776089394115393, −5.92797919688775527211516062738, −5.24812165116164510976879423209, −4.39733659586054030609803456978, −2.57189900470369866196305385215, −1.61055823468542276977837376752, 1.74328030423122966732638982530, 2.83044015013291461450814909337, 3.92180561170235817507739053644, 4.70917791781455737108146808140, 5.28610705581378585643741655866, 7.25391638214641914214231966884, 8.017304479951785897953928513101, 8.626841450569103975034060839486, 10.03785748017199790572649734311, 10.56990916054608998268839616797

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