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Properties

Label 2-666-111.17-c1-0-7
Degree $2$
Conductor $666$
Sign $0.808 + 0.588i$
Analytic cond. $5.31803$
Root an. cond. $2.30608$
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  + (0.573 − 0.819i)2-s + (−0.342 − 0.939i)4-s + (0.980 − 0.0857i)5-s + (1.16 + 0.981i)7-s + (−0.965 − 0.258i)8-s + (0.492 − 0.852i)10-s + (3.02 + 5.24i)11-s + (−1.58 − 0.737i)13-s + (1.47 − 0.395i)14-s + (−0.766 + 0.642i)16-s + (7.17 − 3.34i)17-s + (2.99 − 2.09i)19-s + (−0.415 − 0.892i)20-s + (6.03 + 0.527i)22-s + (−0.211 − 0.788i)23-s + ⋯
L(s)  = 1  + (0.405 − 0.579i)2-s + (−0.171 − 0.469i)4-s + (0.438 − 0.0383i)5-s + (0.442 + 0.370i)7-s + (−0.341 − 0.0915i)8-s + (0.155 − 0.269i)10-s + (0.912 + 1.58i)11-s + (−0.438 − 0.204i)13-s + (0.394 − 0.105i)14-s + (−0.191 + 0.160i)16-s + (1.73 − 0.811i)17-s + (0.686 − 0.480i)19-s + (−0.0930 − 0.199i)20-s + (1.28 + 0.112i)22-s + (−0.0440 − 0.164i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.808 + 0.588i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.808 + 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04042 - 0.664254i\)
\(L(\frac12)\) \(\approx\) \(2.04042 - 0.664254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.573 + 0.819i)T \)
3 \( 1 \)
37 \( 1 + (1.29 + 5.94i)T \)
good5 \( 1 + (-0.980 + 0.0857i)T + (4.92 - 0.868i)T^{2} \)
7 \( 1 + (-1.16 - 0.981i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-3.02 - 5.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.58 + 0.737i)T + (8.35 + 9.95i)T^{2} \)
17 \( 1 + (-7.17 + 3.34i)T + (10.9 - 13.0i)T^{2} \)
19 \( 1 + (-2.99 + 2.09i)T + (6.49 - 17.8i)T^{2} \)
23 \( 1 + (0.211 + 0.788i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.314 - 1.17i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.312 + 0.312i)T - 31iT^{2} \)
41 \( 1 + (8.84 - 3.21i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (4.57 + 4.57i)T + 43iT^{2} \)
47 \( 1 + (-11.5 - 6.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.93 - 9.45i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.977 + 11.1i)T + (-58.1 - 10.2i)T^{2} \)
61 \( 1 + (-3.22 + 6.92i)T + (-39.2 - 46.7i)T^{2} \)
67 \( 1 + (3.99 - 4.75i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (7.37 + 1.30i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 - 8.51iT - 73T^{2} \)
79 \( 1 + (0.00716 + 0.0819i)T + (-77.7 + 13.7i)T^{2} \)
83 \( 1 + (0.0951 - 0.261i)T + (-63.5 - 53.3i)T^{2} \)
89 \( 1 + (9.92 + 0.867i)T + (87.6 + 15.4i)T^{2} \)
97 \( 1 + (7.26 - 1.94i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.25156098253483317755717969806, −9.733227458943813345682160479273, −9.056582602989055659999903884529, −7.68136576834503148931329894350, −6.92214373193147978135385497727, −5.56864172832499023975730233694, −4.98848094677275224843945447663, −3.82930106878101025541759297021, −2.52530826181311822973527144469, −1.42413750434857925482767593992, 1.35372191069369371500901576974, 3.23284629402572855853503366775, 4.02938017678222074196492738033, 5.46750448486327690999220225451, 5.91633879809549021071069685426, 7.01072513874202719944276280990, 7.986448370049164324448739710564, 8.638779169439742551345137869880, 9.763525534743393404384287773540, 10.49696653840477561774229737251

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