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Properties

Label 2-735-105.104-c1-0-64
Degree $2$
Conductor $735$
Sign $0.775 + 0.631i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.775 + 0.631i$
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: \((2,\ 735,\ (\ :1/2),\ 0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.96904 - 1.41276i\)
\(L(\frac12)\) \(\approx\) \(3.96904 - 1.41276i\)
\(L(\frac{3}{2})\) 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

−10.30082396474539816717295219928, −9.377558391835133802781043479608, −8.600555505282312789872151585191, −7.26059236156976883869890613602, −6.62380348195340867763288767265, −5.80801526031067283258949463085, −4.86724506784527821917592013774, −3.77928203320055968719130577201, −2.58980905708593098607955379942, −1.75884279440215119192551067169, 2.10326892888061649709457066339, 3.28985497670938413273958042222, 3.86384376161363932376955317613, 5.03755868609318715677841539470, 5.93783735059647475505870501076, 6.25894709168160370567906906026, 8.072624345284590395829307182957, 8.763077848655248089665746909177, 9.733951660567512724595882228151, 10.57575648322879573686138072810

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