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Properties

Label 2-672-56.3-c1-0-14
Degree $2$
Conductor $672$
Sign $-0.159 + 0.987i$
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  + (0.866 + 0.5i)3-s + (−1.61 − 2.79i)5-s + (1.82 − 1.91i)7-s + (0.499 + 0.866i)9-s + (1.10 − 1.91i)11-s − 5.08·13-s − 3.22i·15-s + (−2.73 − 1.57i)17-s + (−2.93 + 1.69i)19-s + (2.53 − 0.743i)21-s + (2.65 − 1.53i)23-s + (−2.70 + 4.69i)25-s + 0.999i·27-s − 9.88i·29-s + (−1.01 + 1.75i)31-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (−0.721 − 1.25i)5-s + (0.690 − 0.723i)7-s + (0.166 + 0.288i)9-s + (0.333 − 0.577i)11-s − 1.40·13-s − 0.833i·15-s + (−0.663 − 0.383i)17-s + (−0.673 + 0.388i)19-s + (0.554 − 0.162i)21-s + (0.553 − 0.319i)23-s + (−0.541 + 0.938i)25-s + 0.192i·27-s − 1.83i·29-s + (−0.182 + 0.315i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.159 + 0.987i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.872656 - 1.02488i\)
\(L(\frac12)\) \(\approx\) \(0.872656 - 1.02488i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-1.82 + 1.91i)T \)
good5 \( 1 + (1.61 + 2.79i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.10 + 1.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.08T + 13T^{2} \)
17 \( 1 + (2.73 + 1.57i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.93 - 1.69i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.65 + 1.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.88iT - 29T^{2} \)
31 \( 1 + (1.01 - 1.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.798 + 0.460i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.96iT - 41T^{2} \)
43 \( 1 - 6.68T + 43T^{2} \)
47 \( 1 + (-1.06 - 1.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.12 - 1.80i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.6 - 6.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.34 - 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.40 + 7.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + (-7.82 - 4.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.60 - 5.54i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.57iT - 83T^{2} \)
89 \( 1 + (6.32 - 3.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.2iT - 97T^{2} \)
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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

−10.20094012845577752519118491649, −9.234161149595472915275488725507, −8.516772130040945026707905874954, −7.84170204540177237117035318701, −7.01310079169772257469150546898, −5.43768099531230292448406230137, −4.45326996717454895113871991865, −4.04550615406268352824166319079, −2.34624340714043343681091241146, −0.66837890874171920536640087920, 2.06798336969413780908391551939, 2.89866604823846143094518822478, 4.14842420470758862207582084321, 5.21075913774676806229180203957, 6.73108334700652250745133526993, 7.14190334776766716765099117555, 8.055592090037272855102281372910, 8.925641883543172800990404674976, 9.856070339120374869786799475695, 10.91595663483597840868931195662

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