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Properties

Label 2-672-56.37-c1-0-12
Degree $2$
Conductor $672$
Sign $0.894 + 0.447i$
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 + (3.09 + 1.78i)5-s + (−0.993 − 2.45i)7-s + (0.499 − 0.866i)9-s + (0.815 − 0.470i)11-s − 6.15i·13-s + 3.57·15-s + (1.89 + 3.27i)17-s + (−2.09 − 1.20i)19-s + (−2.08 − 1.62i)21-s + (1.49 − 2.58i)23-s + (3.90 + 6.75i)25-s − 0.999i·27-s + 2.68i·29-s + (5.35 + 9.27i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (1.38 + 0.800i)5-s + (−0.375 − 0.926i)7-s + (0.166 − 0.288i)9-s + (0.245 − 0.142i)11-s − 1.70i·13-s + 0.923·15-s + (0.458 + 0.794i)17-s + (−0.479 − 0.276i)19-s + (−0.455 − 0.355i)21-s + (0.311 − 0.539i)23-s + (0.780 + 1.35i)25-s − 0.192i·27-s + 0.497i·29-s + (0.962 + 1.66i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11715 - 0.500641i\)
\(L(\frac12)\) \(\approx\) \(2.11715 - 0.500641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.993 + 2.45i)T \)
good5 \( 1 + (-3.09 - 1.78i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.815 + 0.470i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.15iT - 13T^{2} \)
17 \( 1 + (-1.89 - 3.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.09 + 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.49 + 2.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.68iT - 29T^{2} \)
31 \( 1 + (-5.35 - 9.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.47 + 0.853i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.55T + 41T^{2} \)
43 \( 1 - 3.50iT - 43T^{2} \)
47 \( 1 + (3.42 - 5.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.57 + 3.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.100 - 0.0580i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.06 + 4.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.44 + 1.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.92T + 71T^{2} \)
73 \( 1 + (3.11 + 5.39i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.73 - 4.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.19iT - 83T^{2} \)
89 \( 1 + (-0.910 + 1.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−10.42004884210194017894908128619, −9.814970784218611018536478867634, −8.707926076812787537608120094517, −7.79674810417725292627500908317, −6.73922881011028416216937038006, −6.23436786749440528203978596787, −5.08471995172111105226585604760, −3.47438245572866796061718040886, −2.75738285346487138140664923321, −1.29747183766008296699806750500, 1.72062395210179044886100535524, 2.56619977336183044862398623144, 4.13469875485230426836907162914, 5.15558964443897660010923711992, 5.99663326612935319612093494862, 6.87887114426842719237291561543, 8.300234334383710862064598291130, 9.157099630207811454448027854372, 9.463336108923490212489271202941, 10.14626813000458729908736132780

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