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Properties

Label 2-672-28.19-c3-0-15
Degree $2$
Conductor $672$
Sign $-0.667 - 0.744i$
Analytic cond. $39.6492$
Root an. cond. $6.29676$
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  + (1.5 + 2.59i)3-s + (14.9 + 8.63i)5-s + (15.3 − 10.3i)7-s + (−4.5 + 7.79i)9-s + (−35.5 + 20.5i)11-s − 7.35i·13-s + 51.8i·15-s + (−98.9 + 57.1i)17-s + (−72.7 + 126. i)19-s + (49.9 + 24.2i)21-s + (−181. − 104. i)23-s + (86.5 + 149. i)25-s − 27·27-s + 136.·29-s + (62.3 + 108. i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (1.33 + 0.772i)5-s + (0.827 − 0.561i)7-s + (−0.166 + 0.288i)9-s + (−0.973 + 0.562i)11-s − 0.156i·13-s + 0.891i·15-s + (−1.41 + 0.814i)17-s + (−0.878 + 1.52i)19-s + (0.519 + 0.251i)21-s + (−1.64 − 0.948i)23-s + (0.692 + 1.19i)25-s − 0.192·27-s + 0.873·29-s + (0.361 + 0.625i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.667 - 0.744i$
Analytic conductor: \(39.6492\)
Root analytic conductor: \(6.29676\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :3/2),\ -0.667 - 0.744i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.156999526\)
\(L(\frac12)\) \(\approx\) \(2.156999526\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 - 2.59i)T \)
7 \( 1 + (-15.3 + 10.3i)T \)
good5 \( 1 + (-14.9 - 8.63i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (35.5 - 20.5i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 7.35iT - 2.19e3T^{2} \)
17 \( 1 + (98.9 - 57.1i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (72.7 - 126. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (181. + 104. i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 136.T + 2.43e4T^{2} \)
31 \( 1 + (-62.3 - 108. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (141. - 244. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 208. iT - 6.89e4T^{2} \)
43 \( 1 - 232. iT - 7.95e4T^{2} \)
47 \( 1 + (-318. + 552. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-221. - 382. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (111. + 192. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-436. - 251. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-413. + 238. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 58.6iT - 3.57e5T^{2} \)
73 \( 1 + (-182. + 105. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (25.7 + 14.8i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 516.T + 5.71e5T^{2} \)
89 \( 1 + (-543. - 313. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 609. iT - 9.12e5T^{2} \)
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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

−10.33063457621971528452452024484, −10.02045437664698952931582244988, −8.555398741752235747905681119073, −8.042023383852030357311943024377, −6.72694042209102042408131824478, −6.01184548179763995076726291181, −4.88008923634610482153060932600, −3.98869306352075404710566925531, −2.45535272421425956006560647491, −1.83414782410225819324911478851, 0.51325863833596712369510161543, 2.08462071605653377314358781132, 2.41851029682351621686082657540, 4.44415073757347374383591561453, 5.34706569011045748967792342567, 6.03370664616517620754849802696, 7.14503006330486605084077401923, 8.318322662921837793053120411644, 8.845598413207339173675526862226, 9.546373221625589033067001426952

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