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Properties

Label 2-731-731.208-c1-0-10
Degree $2$
Conductor $731$
Sign $0.339 + 0.940i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
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  − 2.24i·2-s + (−2.58 − 0.692i)3-s − 3.03·4-s + (−2.60 − 0.697i)5-s + (−1.55 + 5.79i)6-s + (2.01 − 0.539i)7-s + 2.31i·8-s + (3.60 + 2.08i)9-s + (−1.56 + 5.83i)10-s + (2.69 + 2.69i)11-s + (7.83 + 2.09i)12-s + (−3.41 + 5.92i)13-s + (−1.20 − 4.51i)14-s + (6.24 + 3.60i)15-s − 0.874·16-s + (3.91 − 1.29i)17-s + ⋯
L(s)  = 1  − 1.58i·2-s + (−1.49 − 0.399i)3-s − 1.51·4-s + (−1.16 − 0.311i)5-s + (−0.634 + 2.36i)6-s + (0.760 − 0.203i)7-s + 0.817i·8-s + (1.20 + 0.693i)9-s + (−0.494 + 1.84i)10-s + (0.813 + 0.813i)11-s + (2.26 + 0.606i)12-s + (−0.948 + 1.64i)13-s + (−0.323 − 1.20i)14-s + (1.61 + 0.930i)15-s − 0.218·16-s + (0.949 − 0.315i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.339 + 0.940i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.339 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.437043 - 0.306974i\)
\(L(\frac12)\) \(\approx\) \(0.437043 - 0.306974i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.91 + 1.29i)T \)
43 \( 1 + (-4.95 + 4.29i)T \)
good2 \( 1 + 2.24iT - 2T^{2} \)
3 \( 1 + (2.58 + 0.692i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (2.60 + 0.697i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.01 + 0.539i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.69 - 2.69i)T + 11iT^{2} \)
13 \( 1 + (3.41 - 5.92i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (-0.0486 + 0.0280i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.368 + 1.37i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.14 - 4.26i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (8.96 + 2.40i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.229 - 0.0614i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.14 + 2.14i)T + 41iT^{2} \)
47 \( 1 - 9.27T + 47T^{2} \)
53 \( 1 + (0.435 - 0.251i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.77iT - 59T^{2} \)
61 \( 1 + (-9.17 + 2.45i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-7.19 - 12.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.876 - 3.27i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.971 + 3.62i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.85 + 0.497i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (12.3 - 7.10i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.10 - 7.11i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.67 + 7.67i)T - 97iT^{2} \)
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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

−10.64438498666495344251038573935, −9.687251189018739519131312348878, −8.837709469141242026434362488857, −7.35955700535069878647843530678, −6.95485214202707167772557342412, −5.31084583950337764069527461764, −4.45342784331901113401685565127, −3.93863506774677775087328248832, −2.01012742777805445015102512876, −0.971437938418006690667775142576, 0.46030009513433041794794474323, 3.56087227038445429483209797611, 4.66002230545654241426316156567, 5.47247194054149826640281778202, 5.92845241309775933824567188319, 7.06419545713801128161403199712, 7.77195834212564199883177003004, 8.358910506465281110913182006919, 9.646434713684302588102301384036, 10.74351232384695721321389055267

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