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Properties

Label 2-735-35.13-c1-0-39
Degree $2$
Conductor $735$
Sign $-0.996 - 0.0891i$
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  + (1.95 − 1.95i)2-s + (0.707 − 0.707i)3-s − 5.67i·4-s + (−2.22 + 0.224i)5-s − 2.76i·6-s + (−7.18 − 7.18i)8-s − 1.00i·9-s + (−3.91 + 4.79i)10-s + 3.09·11-s + (−4.00 − 4.00i)12-s + (1.01 − 1.01i)13-s + (−1.41 + 1.73i)15-s − 16.8·16-s + (2.64 + 2.64i)17-s + (−1.95 − 1.95i)18-s − 2.30·19-s + ⋯
L(s)  = 1  + (1.38 − 1.38i)2-s + (0.408 − 0.408i)3-s − 2.83i·4-s + (−0.994 + 0.100i)5-s − 1.13i·6-s + (−2.54 − 2.54i)8-s − 0.333i·9-s + (−1.23 + 1.51i)10-s + 0.933·11-s + (−1.15 − 1.15i)12-s + (0.280 − 0.280i)13-s + (−0.365 + 0.447i)15-s − 4.20·16-s + (0.641 + 0.641i)17-s + (−0.461 − 0.461i)18-s − 0.527·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.996 - 0.0891i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.996 - 0.0891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130560 + 2.92401i\)
\(L(\frac12)\) \(\approx\) \(0.130560 + 2.92401i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (2.22 - 0.224i)T \)
7 \( 1 \)
good2 \( 1 + (-1.95 + 1.95i)T - 2iT^{2} \)
11 \( 1 - 3.09T + 11T^{2} \)
13 \( 1 + (-1.01 + 1.01i)T - 13iT^{2} \)
17 \( 1 + (-2.64 - 2.64i)T + 17iT^{2} \)
19 \( 1 + 2.30T + 19T^{2} \)
23 \( 1 + (1.14 + 1.14i)T + 23iT^{2} \)
29 \( 1 - 5.62iT - 29T^{2} \)
31 \( 1 - 0.287iT - 31T^{2} \)
37 \( 1 + (-1.96 + 1.96i)T - 37iT^{2} \)
41 \( 1 + 10.7iT - 41T^{2} \)
43 \( 1 + (1.32 + 1.32i)T + 43iT^{2} \)
47 \( 1 + (-2.72 - 2.72i)T + 47iT^{2} \)
53 \( 1 + (-2.49 - 2.49i)T + 53iT^{2} \)
59 \( 1 - 8.98T + 59T^{2} \)
61 \( 1 + 8.64iT - 61T^{2} \)
67 \( 1 + (-8.73 + 8.73i)T - 67iT^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + (-1.16 + 1.16i)T - 73iT^{2} \)
79 \( 1 - 7.82iT - 79T^{2} \)
83 \( 1 + (2.41 - 2.41i)T - 83iT^{2} \)
89 \( 1 + 9.44T + 89T^{2} \)
97 \( 1 + (8.04 + 8.04i)T + 97iT^{2} \)
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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

−10.38573937475140033086740896155, −9.294489631442924797327439799965, −8.394947277693773978329454389446, −7.03223040049935310548696591013, −6.19307123744491183698639198709, −5.08900791962491905176650034716, −3.86134489005081004223368905620, −3.59295275686299131252222271399, −2.28084443172177351251572547153, −0.994191196237968258314802429054, 2.86986847691468192248752939331, 3.89546048480594595910184967544, 4.32780356650236383023459001724, 5.37103185978541244037595420171, 6.44100003071461804755535742173, 7.19592687062798055051613366806, 8.075790151072450720181950926064, 8.600454009892828250238545588771, 9.660442526230551950346919152739, 11.40396464186381418472432173207

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