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Properties

Label 2-735-5.4-c1-0-26
Degree $2$
Conductor $735$
Sign $0.749 + 0.662i$
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  + 0.193i·2-s − i·3-s + 1.96·4-s + (1.48 − 1.67i)5-s + 0.193·6-s + 0.768i·8-s − 9-s + (0.324 + 0.287i)10-s + 2·11-s − 1.96i·12-s + 1.35i·13-s + (−1.67 − 1.48i)15-s + 3.77·16-s + 3.35i·17-s − 0.193i·18-s + 5.35·19-s + ⋯
L(s)  = 1  + 0.137i·2-s − 0.577i·3-s + 0.981·4-s + (0.662 − 0.749i)5-s + 0.0791·6-s + 0.271i·8-s − 0.333·9-s + (0.102 + 0.0908i)10-s + 0.603·11-s − 0.566i·12-s + 0.374i·13-s + (−0.432 − 0.382i)15-s + 0.943·16-s + 0.812i·17-s − 0.0457i·18-s + 1.22·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05426 - 0.777960i\)
\(L(\frac12)\) \(\approx\) \(2.05426 - 0.777960i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-1.48 + 1.67i)T \)
7 \( 1 \)
good2 \( 1 - 0.193iT - 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
17 \( 1 - 3.35iT - 17T^{2} \)
19 \( 1 - 5.35T + 19T^{2} \)
23 \( 1 + 4.96iT - 23T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 + 4.57T + 31T^{2} \)
37 \( 1 + 0.775iT - 37T^{2} \)
41 \( 1 + 3.73T + 41T^{2} \)
43 \( 1 - 12.6iT - 43T^{2} \)
47 \( 1 + 9.92iT - 47T^{2} \)
53 \( 1 + 8.57iT - 53T^{2} \)
59 \( 1 + 8.62T + 59T^{2} \)
61 \( 1 - 8.70T + 61T^{2} \)
67 \( 1 - 9.92iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 9.35iT - 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 3.22iT - 83T^{2} \)
89 \( 1 - 1.03T + 89T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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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

−10.26884721138839601965819089239, −9.374095043172581289989331617117, −8.499420311423106930490471755231, −7.60418091438421953162193538641, −6.69235363371128581230220093425, −5.99995695299882157037346483582, −5.13050816174725436104763815235, −3.63547783436169013411225716614, −2.20685070228205092138121302023, −1.33517352106343975310830010203, 1.67309995943286332352979678029, 2.93781293254516387576425200299, 3.65395471218706518316633281896, 5.34936345412940930464020021887, 5.93433136360596234907673704135, 7.10564120078166246438989794012, 7.55281737178351559895642181099, 9.145434762624882563075250675769, 9.702723813767451379009416699077, 10.51062732391119466601484402749

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