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Properties

Label 2-672-224.205-c1-0-42
Degree $2$
Conductor $672$
Sign $0.702 + 0.711i$
Analytic cond. $5.36594$
Root an. cond. $2.31645$
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  + (1.27 − 0.622i)2-s + (−0.991 + 0.130i)3-s + (1.22 − 1.58i)4-s + (−0.309 + 2.35i)5-s + (−1.17 + 0.782i)6-s + (1.14 − 2.38i)7-s + (0.573 − 2.76i)8-s + (0.965 − 0.258i)9-s + (1.07 + 3.18i)10-s + (0.501 − 0.653i)11-s + (−1.00 + 1.72i)12-s + (3.37 + 1.39i)13-s + (−0.0241 − 3.74i)14-s − 2.37i·15-s + (−0.994 − 3.87i)16-s + (−4.41 + 2.54i)17-s + ⋯
L(s)  = 1  + (0.898 − 0.439i)2-s + (−0.572 + 0.0753i)3-s + (0.612 − 0.790i)4-s + (−0.138 + 1.05i)5-s + (−0.480 + 0.319i)6-s + (0.434 − 0.900i)7-s + (0.202 − 0.979i)8-s + (0.321 − 0.0862i)9-s + (0.338 + 1.00i)10-s + (0.151 − 0.196i)11-s + (−0.291 + 0.498i)12-s + (0.936 + 0.387i)13-s + (−0.00646 − 0.999i)14-s − 0.612i·15-s + (−0.248 − 0.968i)16-s + (−1.07 + 0.617i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11593 - 0.884164i\)
\(L(\frac12)\) \(\approx\) \(2.11593 - 0.884164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 + 0.622i)T \)
3 \( 1 + (0.991 - 0.130i)T \)
7 \( 1 + (-1.14 + 2.38i)T \)
good5 \( 1 + (0.309 - 2.35i)T + (-4.82 - 1.29i)T^{2} \)
11 \( 1 + (-0.501 + 0.653i)T + (-2.84 - 10.6i)T^{2} \)
13 \( 1 + (-3.37 - 1.39i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + (4.41 - 2.54i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.10 + 4.68i)T + (4.91 - 18.3i)T^{2} \)
23 \( 1 + (-3.19 + 0.856i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.19 + 2.88i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-2.94 - 5.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.873 + 6.63i)T + (-35.7 - 9.57i)T^{2} \)
41 \( 1 + (4.87 + 4.87i)T + 41iT^{2} \)
43 \( 1 + (-2.84 - 6.86i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (5.87 + 3.39i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.04 - 3.97i)T + (-13.7 - 51.1i)T^{2} \)
59 \( 1 + (-6.76 - 5.18i)T + (15.2 + 56.9i)T^{2} \)
61 \( 1 + (-6.90 - 8.99i)T + (-15.7 + 58.9i)T^{2} \)
67 \( 1 + (8.67 - 1.14i)T + (64.7 - 17.3i)T^{2} \)
71 \( 1 + (6.86 - 6.86i)T - 71iT^{2} \)
73 \( 1 + (1.91 - 7.15i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (14.0 + 8.10i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.35 + 0.562i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-1.79 - 6.71i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−10.75729121479448711241176551732, −10.03452443724182425410395747555, −8.767764902796554793543810186325, −7.16399544444224891014498725493, −6.86171138796226391394863242292, −5.86323371714330423148958846929, −4.70596564024355952084653997286, −3.88666007825032873597718742949, −2.83971910735597748660954673445, −1.20837131023724046093624248848, 1.52443082647850172394811198824, 3.15304134864518083709540246941, 4.51489672876644823167858407075, 5.15622134902396474142955874496, 5.87924104947598934991393056024, 6.85325954879845033749385648045, 8.033331034055632344754135123370, 8.598941281547003842120875805378, 9.642942380689385657099477042014, 11.07881234681136560151488458061

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