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Properties

Label 2-672-224.195-c1-0-9
Degree $2$
Conductor $672$
Sign $-0.648 - 0.761i$
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  + (−1.37 + 0.345i)2-s + (0.923 + 0.382i)3-s + (1.76 − 0.947i)4-s + (−0.737 + 0.305i)5-s + (−1.39 − 0.205i)6-s + (−2.64 + 0.158i)7-s + (−2.08 + 1.90i)8-s + (0.707 + 0.707i)9-s + (0.906 − 0.674i)10-s + (−0.466 − 1.12i)11-s + (1.98 − 0.201i)12-s + (5.51 + 2.28i)13-s + (3.56 − 1.12i)14-s − 0.798·15-s + (2.20 − 3.33i)16-s − 3.98·17-s + ⋯
L(s)  = 1  + (−0.969 + 0.244i)2-s + (0.533 + 0.220i)3-s + (0.880 − 0.473i)4-s + (−0.329 + 0.136i)5-s + (−0.571 − 0.0838i)6-s + (−0.998 + 0.0598i)7-s + (−0.738 + 0.674i)8-s + (0.235 + 0.235i)9-s + (0.286 − 0.213i)10-s + (−0.140 − 0.339i)11-s + (0.574 − 0.0582i)12-s + (1.52 + 0.633i)13-s + (0.953 − 0.301i)14-s − 0.206·15-s + (0.550 − 0.834i)16-s − 0.965·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.648 - 0.761i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.648 - 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.277285 + 0.600150i\)
\(L(\frac12)\) \(\approx\) \(0.277285 + 0.600150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.345i)T \)
3 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (2.64 - 0.158i)T \)
good5 \( 1 + (0.737 - 0.305i)T + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (0.466 + 1.12i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-5.51 - 2.28i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 3.98T + 17T^{2} \)
19 \( 1 + (1.67 - 4.03i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.31 - 5.31i)T - 23iT^{2} \)
29 \( 1 + (-2.49 - 1.03i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 2.75T + 31T^{2} \)
37 \( 1 + (-2.06 - 4.98i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.12 + 1.12i)T - 41iT^{2} \)
43 \( 1 + (4.65 + 11.2i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 8.41iT - 47T^{2} \)
53 \( 1 + (12.9 - 5.38i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.99 - 4.81i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (5.01 - 12.1i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-0.270 + 0.654i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (0.958 + 0.958i)T + 71iT^{2} \)
73 \( 1 + (-7.21 + 7.21i)T - 73iT^{2} \)
79 \( 1 + 3.79T + 79T^{2} \)
83 \( 1 + (0.815 - 1.96i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (0.266 + 0.266i)T + 89iT^{2} \)
97 \( 1 - 4.05iT - 97T^{2} \)
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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

−10.60833661834660913075560527487, −9.805973778908959786568491063400, −9.021728300910859254596274365528, −8.384097272809532285116412722503, −7.51357309512034707038512520316, −6.43417521237071314595915181230, −5.88391501573267999129858862542, −4.03670806559279087401277130904, −3.12988171837052893404719345368, −1.68525054419926592097306824074, 0.43731813791760001646896298233, 2.18608188384702248876531297712, 3.24260469769798888388072103871, 4.26602329022302401915912304759, 6.30730382666980480777294005156, 6.59836716067356652709399713669, 7.967854587744791679163484694054, 8.401538257159076106775209443636, 9.289257086303265841695351535185, 10.07816771032775148053569423105

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