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Properties

Label 2-735-49.29-c1-0-31
Degree $2$
Conductor $735$
Sign $-0.400 - 0.916i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
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  + (−0.980 − 0.472i)2-s + (−0.222 − 0.974i)3-s + (−0.508 − 0.638i)4-s + (0.222 + 0.974i)5-s + (−0.242 + 1.06i)6-s + (−0.163 − 2.64i)7-s + (0.681 + 2.98i)8-s + (−0.900 + 0.433i)9-s + (0.242 − 1.06i)10-s + (−0.505 − 0.243i)11-s + (−0.508 + 0.638i)12-s + (−2.00 − 0.964i)13-s + (−1.08 + 2.66i)14-s + (0.900 − 0.433i)15-s + (0.378 − 1.65i)16-s + (0.691 − 0.867i)17-s + ⋯
L(s)  = 1  + (−0.693 − 0.333i)2-s + (−0.128 − 0.562i)3-s + (−0.254 − 0.319i)4-s + (0.0995 + 0.436i)5-s + (−0.0988 + 0.433i)6-s + (−0.0618 − 0.998i)7-s + (0.241 + 1.05i)8-s + (−0.300 + 0.144i)9-s + (0.0765 − 0.335i)10-s + (−0.152 − 0.0734i)11-s + (−0.146 + 0.184i)12-s + (−0.555 − 0.267i)13-s + (−0.290 + 0.712i)14-s + (0.232 − 0.112i)15-s + (0.0946 − 0.414i)16-s + (0.167 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.400 - 0.916i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.400 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0342336 + 0.0523398i\)
\(L(\frac12)\) \(\approx\) \(0.0342336 + 0.0523398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (0.163 + 2.64i)T \)
good2 \( 1 + (0.980 + 0.472i)T + (1.24 + 1.56i)T^{2} \)
11 \( 1 + (0.505 + 0.243i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (2.00 + 0.964i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-0.691 + 0.867i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + 5.66T + 19T^{2} \)
23 \( 1 + (1.36 + 1.70i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (4.32 - 5.42i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 - 1.38T + 31T^{2} \)
37 \( 1 + (3.00 - 3.76i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (0.572 + 2.50i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (1.14 - 5.02i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-4.94 - 2.38i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (0.147 + 0.184i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (1.09 - 4.79i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (8.16 - 10.2i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 - 6.69T + 67T^{2} \)
71 \( 1 + (-0.583 - 0.731i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.32 + 0.638i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + 8.13T + 79T^{2} \)
83 \( 1 + (-6.30 + 3.03i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (4.25 - 2.04i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + 2.55T + 97T^{2} \)
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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

−9.994330454850618886173975258992, −8.992061413719642216808797481577, −8.087779336119611114412416650589, −7.30745321288948615759283306299, −6.41702140095626844694714179309, −5.33783373487406958090327103877, −4.24420744287188771294770339589, −2.73302862746738635758818850436, −1.48048179064355985301845045656, −0.04146820945821138515735851006, 2.17702746569796648729947235096, 3.68507011287851952761139805565, 4.63929499320423472720863808215, 5.65023920887033459162969260365, 6.63105003691423400904709062168, 7.82863908097138621198048529901, 8.509375249879492678941784397711, 9.253733324999627316429727133071, 9.767823991305934420344344081820, 10.72094101741974410946925156574

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