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Properties

Label 2-735-105.104-c1-0-55
Degree $2$
Conductor $735$
Sign $0.577 + 0.816i$
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  + 2.20·2-s + (−1.59 + 0.672i)3-s + 2.87·4-s + (−1.33 − 1.79i)5-s + (−3.52 + 1.48i)6-s + 1.94·8-s + (2.09 − 2.14i)9-s + (−2.94 − 3.96i)10-s − 3.81i·11-s + (−4.59 + 1.93i)12-s + 6.50·13-s + (3.33 + 1.97i)15-s − 1.46·16-s − 2.94i·17-s + (4.63 − 4.74i)18-s − 2.03i·19-s + ⋯
L(s)  = 1  + 1.56·2-s + (−0.921 + 0.388i)3-s + 1.43·4-s + (−0.595 − 0.803i)5-s + (−1.43 + 0.606i)6-s + 0.687·8-s + (0.698 − 0.715i)9-s + (−0.930 − 1.25i)10-s − 1.15i·11-s + (−1.32 + 0.558i)12-s + 1.80·13-s + (0.860 + 0.509i)15-s − 0.366·16-s − 0.713i·17-s + (1.09 − 1.11i)18-s − 0.466i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.577 + 0.816i$
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: \((2,\ 735,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11623 - 1.09573i\)
\(L(\frac12)\) \(\approx\) \(2.11623 - 1.09573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.59 - 0.672i)T \)
5 \( 1 + (1.33 + 1.79i)T \)
7 \( 1 \)
good2 \( 1 - 2.20T + 2T^{2} \)
11 \( 1 + 3.81iT - 11T^{2} \)
13 \( 1 - 6.50T + 13T^{2} \)
17 \( 1 + 2.94iT - 17T^{2} \)
19 \( 1 + 2.03iT - 19T^{2} \)
23 \( 1 - 3.00T + 23T^{2} \)
29 \( 1 + 2.25iT - 29T^{2} \)
31 \( 1 + 6.66iT - 31T^{2} \)
37 \( 1 + 3.36iT - 37T^{2} \)
41 \( 1 - 3.51T + 41T^{2} \)
43 \( 1 - 7.03iT - 43T^{2} \)
47 \( 1 - 5.01iT - 47T^{2} \)
53 \( 1 - 1.93T + 53T^{2} \)
59 \( 1 + 7.93T + 59T^{2} \)
61 \( 1 - 13.6iT - 61T^{2} \)
67 \( 1 - 10.7iT - 67T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 3.30T + 73T^{2} \)
79 \( 1 + 5.84T + 79T^{2} \)
83 \( 1 - 2.66iT - 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 - 4.46T + 97T^{2} \)
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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

−10.93301843894301442919077356085, −9.355460695632235591896599177010, −8.599170944962777039046427249857, −7.30810930191706173209747394489, −6.06985054164498159304404563738, −5.77995876747166557751064803975, −4.69328028638037637133445304635, −4.03514697579625991412132295654, −3.16446787402903989200857722946, −0.894763379409199846864302842472, 1.78552950119263528498553612313, 3.33011865316252625274298729232, 4.12669709546072409014480986369, 5.06844404417235034972374873041, 6.08691041186726036232524374500, 6.65269285952482635772693854377, 7.39085578740451005963563014135, 8.573732931158506831992676325675, 10.20243078169684753873470959236, 10.93383822154422195346996978985

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