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Properties

Label 2-735-105.104-c1-0-9
Degree $2$
Conductor $735$
Sign $-0.932 - 0.361i$
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  + 2.04·2-s + (−0.965 + 1.43i)3-s + 2.18·4-s + (−2.23 + 0.144i)5-s + (−1.97 + 2.94i)6-s + 0.376·8-s + (−1.13 − 2.77i)9-s + (−4.56 + 0.294i)10-s + 5.15i·11-s + (−2.10 + 3.14i)12-s − 2.98·13-s + (1.94 − 3.34i)15-s − 3.59·16-s + 1.35i·17-s + (−2.32 − 5.67i)18-s + 3.09i·19-s + ⋯
L(s)  = 1  + 1.44·2-s + (−0.557 + 0.830i)3-s + 1.09·4-s + (−0.997 + 0.0644i)5-s + (−0.806 + 1.20i)6-s + 0.133·8-s + (−0.378 − 0.925i)9-s + (−1.44 + 0.0931i)10-s + 1.55i·11-s + (−0.608 + 0.906i)12-s − 0.826·13-s + (0.502 − 0.864i)15-s − 0.899·16-s + 0.329i·17-s + (−0.547 − 1.33i)18-s + 0.709i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.932 - 0.361i$
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: \((2,\ 735,\ (\ :1/2),\ -0.932 - 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218239 + 1.16775i\)
\(L(\frac12)\) \(\approx\) \(0.218239 + 1.16775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 - 1.43i)T \)
5 \( 1 + (2.23 - 0.144i)T \)
7 \( 1 \)
good2 \( 1 - 2.04T + 2T^{2} \)
11 \( 1 - 5.15iT - 11T^{2} \)
13 \( 1 + 2.98T + 13T^{2} \)
17 \( 1 - 1.35iT - 17T^{2} \)
19 \( 1 - 3.09iT - 19T^{2} \)
23 \( 1 + 7.22T + 23T^{2} \)
29 \( 1 - 2.69iT - 29T^{2} \)
31 \( 1 + 4.35iT - 31T^{2} \)
37 \( 1 + 7.59iT - 37T^{2} \)
41 \( 1 - 7.68T + 41T^{2} \)
43 \( 1 - 7.79iT - 43T^{2} \)
47 \( 1 - 6.07iT - 47T^{2} \)
53 \( 1 - 6.29T + 53T^{2} \)
59 \( 1 - 4.25T + 59T^{2} \)
61 \( 1 + 5.21iT - 61T^{2} \)
67 \( 1 - 5.65iT - 67T^{2} \)
71 \( 1 - 11.2iT - 71T^{2} \)
73 \( 1 + 4.91T + 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 - 12.1iT - 83T^{2} \)
89 \( 1 - 9.32T + 89T^{2} \)
97 \( 1 + 19.2T + 97T^{2} \)
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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

−11.03585699026048168274601740514, −10.05640423981819176372462655456, −9.318069426111412856597449956954, −7.923304624342584577065927857286, −7.02309751085150498187544274717, −6.01854433484917196507362020438, −5.10693927932161645706507340204, −4.20926688814946893697454446723, −3.93382510392389144054828668998, −2.49971431048339394609921738520, 0.38500251701450582364588808162, 2.50995963672074076059337696811, 3.52409678387261966683312433786, 4.61254020082505631431237390623, 5.44138138762383251130307496389, 6.27645556575646258174238603500, 7.11127258379351833464148964884, 8.017914062218908889111728640358, 8.878590637466473979369732016908, 10.49406404315427959387973212450

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