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Properties

Label 2-672-12.11-c1-0-8
Degree $2$
Conductor $672$
Sign $0.356 - 0.934i$
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  + (0.707 + 1.58i)3-s − 1.41i·5-s + i·7-s + (−2.00 + 2.23i)9-s + 4.47·11-s − 0.837·13-s + (2.23 − 1.00i)15-s + 1.64i·17-s + 7.16i·19-s + (−1.58 + 0.707i)21-s + 5.65·23-s + 2.99·25-s + (−4.94 − 1.58i)27-s + 7.30i·29-s − 6.32i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.912i)3-s − 0.632i·5-s + 0.377i·7-s + (−0.666 + 0.745i)9-s + 1.34·11-s − 0.232·13-s + (0.577 − 0.258i)15-s + 0.398i·17-s + 1.64i·19-s + (−0.345 + 0.154i)21-s + 1.17·23-s + 0.599·25-s + (−0.952 − 0.304i)27-s + 1.35i·29-s − 1.13i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.356 - 0.934i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43951 + 0.991107i\)
\(L(\frac12)\) \(\approx\) \(1.43951 + 0.991107i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 1.58i)T \)
7 \( 1 - iT \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 - 4.47T + 11T^{2} \)
13 \( 1 + 0.837T + 13T^{2} \)
17 \( 1 - 1.64iT - 17T^{2} \)
19 \( 1 - 7.16iT - 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 - 7.30iT - 29T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 + 8.32T + 37T^{2} \)
41 \( 1 + 1.18iT - 41T^{2} \)
43 \( 1 - 4.32iT - 43T^{2} \)
47 \( 1 - 8.94T + 47T^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 + 3.16T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 1.18T + 71T^{2} \)
73 \( 1 + 8.32T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−10.53251811413264089749985918418, −9.694621618537934619245263573856, −8.883690308461253352283844103832, −8.499647663025469116498515056363, −7.24068265789125074127477034451, −5.99339349753668320810140090517, −5.09922228460727550506679782017, −4.12690368845679156928238890074, −3.24175488480860312762508301777, −1.63114332403665162694278117867, 1.00223227001158987024373141102, 2.51725997783230688746864151443, 3.46065635417491917337146806182, 4.77848505323463176883509018189, 6.21666417321056354989205606381, 7.02158452246868418936553174074, 7.32608449502901790593179772176, 8.822952262189913924124742479556, 9.128863025577550589625023657202, 10.43169577550765251398237465414

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