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Properties

Label 2-735-49.15-c1-0-30
Degree $2$
Conductor $735$
Sign $-0.253 + 0.967i$
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  + (0.189 − 0.831i)2-s + (0.623 − 0.781i)3-s + (1.14 + 0.552i)4-s + (0.623 − 0.781i)5-s + (−0.531 − 0.666i)6-s + (−2.46 − 0.952i)7-s + (1.73 − 2.18i)8-s + (−0.222 − 0.974i)9-s + (−0.531 − 0.666i)10-s + (−0.625 + 2.74i)11-s + (1.14 − 0.552i)12-s + (1.23 − 5.42i)13-s + (−1.26 + 1.87i)14-s + (−0.222 − 0.974i)15-s + (0.104 + 0.131i)16-s + (3.87 − 1.86i)17-s + ⋯
L(s)  = 1  + (0.134 − 0.587i)2-s + (0.359 − 0.451i)3-s + (0.573 + 0.276i)4-s + (0.278 − 0.349i)5-s + (−0.216 − 0.272i)6-s + (−0.932 − 0.360i)7-s + (0.615 − 0.771i)8-s + (−0.0741 − 0.324i)9-s + (−0.168 − 0.210i)10-s + (−0.188 + 0.826i)11-s + (0.331 − 0.159i)12-s + (0.343 − 1.50i)13-s + (−0.336 + 0.499i)14-s + (−0.0574 − 0.251i)15-s + (0.0262 + 0.0329i)16-s + (0.939 − 0.452i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.253 + 0.967i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.253 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27190 - 1.64776i\)
\(L(\frac12)\) \(\approx\) \(1.27190 - 1.64776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + (2.46 + 0.952i)T \)
good2 \( 1 + (-0.189 + 0.831i)T + (-1.80 - 0.867i)T^{2} \)
11 \( 1 + (0.625 - 2.74i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-1.23 + 5.42i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-3.87 + 1.86i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + 1.56T + 19T^{2} \)
23 \( 1 + (5.71 + 2.75i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (2.35 - 1.13i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 - 8.87T + 31T^{2} \)
37 \( 1 + (-4.72 + 2.27i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (1.73 - 2.17i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-2.68 - 3.36i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (2.58 - 11.3i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (0.879 + 0.423i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (3.43 + 4.31i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-10.8 + 5.24i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + 8.36T + 67T^{2} \)
71 \( 1 + (4.72 + 2.27i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.77 - 12.1i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + 5.88T + 79T^{2} \)
83 \( 1 + (-1.75 - 7.68i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-0.778 - 3.41i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + 6.73T + 97T^{2} \)
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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

−10.02350533997243749660197059186, −9.693735448004402854660477475604, −8.162225216175250773152090144661, −7.66796769043208589847850381642, −6.63135071073122320509805375356, −5.84472492984896821317880261178, −4.35478732329581819172033272306, −3.22884404606525913981400223254, −2.47774080176655663271990301530, −1.01611333791507798116387747777, 1.94013692635664205687647030201, 3.08000928147509741993829871313, 4.17263051863819818109577888555, 5.64464759050625587358327558849, 6.14572863255748686633224979245, 6.95124254242505271649228010488, 8.037773172549761441239133036335, 8.870405830364642140702078734342, 9.903055260266149774441928372006, 10.37790443977387496623566460001

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