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Properties

Label 2-1512-21.17-c1-0-24
Degree $2$
Conductor $1512$
Sign $-0.261 + 0.965i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (−1.12 − 1.95i)5-s + (2.62 − 0.349i)7-s + (−2.61 − 1.50i)11-s + 3.29i·13-s + (3.31 − 5.73i)17-s + (−0.605 + 0.349i)19-s + (1.73 − i)23-s + (−0.0520 + 0.0902i)25-s + 1.45i·29-s + (0.323 + 0.186i)31-s + (−3.64 − 4.73i)35-s + (−5.63 − 9.75i)37-s − 2.10·41-s − 43-s + (−3.06 − 5.30i)47-s + ⋯
L(s)  = 1  + (−0.505 − 0.875i)5-s + (0.991 − 0.132i)7-s + (−0.787 − 0.454i)11-s + 0.913i·13-s + (0.803 − 1.39i)17-s + (−0.138 + 0.0801i)19-s + (0.361 − 0.208i)23-s + (−0.0104 + 0.0180i)25-s + 0.270i·29-s + (0.0580 + 0.0335i)31-s + (−0.616 − 0.800i)35-s + (−0.925 − 1.60i)37-s − 0.328·41-s − 0.152·43-s + (−0.446 − 0.773i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.261 + 0.965i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.261 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.350396480\)
\(L(\frac12)\) \(\approx\) \(1.350396480\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.62 + 0.349i)T \)
good5 \( 1 + (1.12 + 1.95i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.61 + 1.50i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.29iT - 13T^{2} \)
17 \( 1 + (-3.31 + 5.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.605 - 0.349i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.45iT - 29T^{2} \)
31 \( 1 + (-0.323 - 0.186i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.63 + 9.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.10T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (3.06 + 5.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0299 + 0.0173i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.61 - 4.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.36 - 2.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.34 + 2.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (11.7 + 6.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.52 - 13.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.58T + 83T^{2} \)
89 \( 1 + (7.50 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.09iT - 97T^{2} \)
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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

−8.959449413188930618248781823119, −8.559873189220197939703644039925, −7.64359869397836609102769920587, −7.09870257394006704013261215559, −5.70733495156937855776799169167, −4.96323454128858322947070149510, −4.38852000963870285184725583903, −3.17777457596743787760662344427, −1.85137024260003182634226210711, −0.55142050024298411466691104180, 1.50100162175444323084200927107, 2.77728962416655073747544244017, 3.61356882510127213520200859461, 4.77044444982280116020478692238, 5.52157940096113223088287085705, 6.51737260826467438102783418485, 7.54349808816161589226465207238, 7.957919253563786325469736642871, 8.657572058512246247060385310108, 10.01274939022891581126960329242

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