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Properties

Label 2-735-105.32-c1-0-31
Degree $2$
Conductor $735$
Sign $0.815 - 0.578i$
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  + (0.298 + 0.0799i)2-s + (1.64 + 0.540i)3-s + (−1.64 − 0.952i)4-s + (1.56 + 1.59i)5-s + (0.447 + 0.292i)6-s + (−0.852 − 0.852i)8-s + (2.41 + 1.77i)9-s + (0.340 + 0.600i)10-s + (0.660 + 0.381i)11-s + (−2.19 − 2.45i)12-s + (2.27 − 2.27i)13-s + (1.71 + 3.47i)15-s + (1.71 + 2.97i)16-s + (1.25 + 4.69i)17-s + (0.578 + 0.723i)18-s + (−1.41 + 0.818i)19-s + ⋯
L(s)  = 1  + (0.210 + 0.0565i)2-s + (0.950 + 0.312i)3-s + (−0.824 − 0.476i)4-s + (0.701 + 0.712i)5-s + (0.182 + 0.119i)6-s + (−0.301 − 0.301i)8-s + (0.805 + 0.593i)9-s + (0.107 + 0.189i)10-s + (0.199 + 0.114i)11-s + (−0.634 − 0.709i)12-s + (0.629 − 0.629i)13-s + (0.443 + 0.896i)15-s + (0.429 + 0.744i)16-s + (0.305 + 1.13i)17-s + (0.136 + 0.170i)18-s + (−0.325 + 0.187i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.815 - 0.578i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.815 - 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16019 + 0.687830i\)
\(L(\frac12)\) \(\approx\) \(2.16019 + 0.687830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 - 0.540i)T \)
5 \( 1 + (-1.56 - 1.59i)T \)
7 \( 1 \)
good2 \( 1 + (-0.298 - 0.0799i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-0.660 - 0.381i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.27 + 2.27i)T - 13iT^{2} \)
17 \( 1 + (-1.25 - 4.69i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.41 - 0.818i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.98 + 7.39i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 + (2.96 - 5.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.915 - 3.41i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.35iT - 41T^{2} \)
43 \( 1 + (-2.69 + 2.69i)T - 43iT^{2} \)
47 \( 1 + (4.14 + 1.10i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.71 - 1.79i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.84 + 6.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.19 - 3.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0471 - 0.0126i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (0.359 + 1.34i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.66 - 2.11i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.05 + 5.05i)T + 83iT^{2} \)
89 \( 1 + (0.453 + 0.785i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.73 + 3.73i)T + 97iT^{2} \)
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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

−10.49333633944720333833458783051, −9.654432850313757827842487364584, −8.723883021541388706676987759275, −8.231374590533811560651143389116, −6.83728591214446221948383645336, −6.01958631987273528093229370008, −4.95707601380947380795947613393, −3.90482473199474181240144507172, −2.99679056180902834437501839009, −1.58686342186591413262986443750, 1.22159194608916179162961701852, 2.66478475176719440370225909803, 3.77635311552674073577808065782, 4.67656865662514638013301225485, 5.70956745044771929142097323707, 6.92786741087194099411731124492, 7.947028194258512866276347477630, 8.673586977345201354295754917244, 9.436447137566165085592782372225, 9.694755526331229235547518376373

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