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Properties

Label 2-735-105.104-c1-0-27
Degree $2$
Conductor $735$
Sign $0.207 - 0.978i$
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  − 2.50·2-s + (1.47 + 0.900i)3-s + 4.28·4-s + (1.94 + 1.11i)5-s + (−3.71 − 2.25i)6-s − 5.74·8-s + (1.37 + 2.66i)9-s + (−4.86 − 2.78i)10-s − 1.71i·11-s + (6.34 + 3.86i)12-s − 0.360·13-s + (1.87 + 3.39i)15-s + 5.82·16-s + 2.54i·17-s + (−3.45 − 6.68i)18-s + 5.69i·19-s + ⋯
L(s)  = 1  − 1.77·2-s + (0.854 + 0.519i)3-s + 2.14·4-s + (0.867 + 0.496i)5-s + (−1.51 − 0.922i)6-s − 2.03·8-s + (0.459 + 0.888i)9-s + (−1.53 − 0.880i)10-s − 0.518i·11-s + (1.83 + 1.11i)12-s − 0.100·13-s + (0.483 + 0.875i)15-s + 1.45·16-s + 0.616i·17-s + (−0.814 − 1.57i)18-s + 1.30i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.207 - 0.978i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.207 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.809841 + 0.655838i\)
\(L(\frac12)\) \(\approx\) \(0.809841 + 0.655838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.47 - 0.900i)T \)
5 \( 1 + (-1.94 - 1.11i)T \)
7 \( 1 \)
good2 \( 1 + 2.50T + 2T^{2} \)
11 \( 1 + 1.71iT - 11T^{2} \)
13 \( 1 + 0.360T + 13T^{2} \)
17 \( 1 - 2.54iT - 17T^{2} \)
19 \( 1 - 5.69iT - 19T^{2} \)
23 \( 1 - 2.50T + 23T^{2} \)
29 \( 1 + 3.76iT - 29T^{2} \)
31 \( 1 - 2.78iT - 31T^{2} \)
37 \( 1 - 3.30iT - 37T^{2} \)
41 \( 1 + 2.63T + 41T^{2} \)
43 \( 1 + 10.0iT - 43T^{2} \)
47 \( 1 + 5.82iT - 47T^{2} \)
53 \( 1 + 1.45T + 53T^{2} \)
59 \( 1 - 6.85T + 59T^{2} \)
61 \( 1 - 1.60iT - 61T^{2} \)
67 \( 1 - 2.49iT - 67T^{2} \)
71 \( 1 - 13.1iT - 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 + 3.50iT - 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 - 8.18T + 97T^{2} \)
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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

−10.22087521721293017209735631738, −9.788673809658930161462640918794, −8.823676198003900607897496753066, −8.354455648337738125860754933255, −7.42179004434072759182401867049, −6.54127754642121323892632152581, −5.46463616526321950459683052746, −3.65063820499156180419473575499, −2.52370284587940326639470499719, −1.57026527668654308302546669689, 0.908075035996219972006883584856, 2.02238844031210982224462403928, 2.87505531776021875232724394232, 4.81755068285948398490261584676, 6.32836566668318328835859510878, 7.04571109343801880813484377756, 7.79560624218774202117901710137, 8.708789503231142456048972090184, 9.328229611772531500742135791228, 9.679054513160821541164332301262

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