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Properties

Label 2-735-15.8-c1-0-19
Degree $2$
Conductor $735$
Sign $0.991 + 0.131i$
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  + (−1.06 − 1.06i)2-s + (1.73 + 0.0150i)3-s + 0.285i·4-s + (−2.03 + 0.933i)5-s + (−1.83 − 1.86i)6-s + (−1.83 + 1.83i)8-s + (2.99 + 0.0520i)9-s + (3.16 + 1.17i)10-s + 0.914i·11-s + (−0.00428 + 0.493i)12-s + (3.07 + 3.07i)13-s + (−3.53 + 1.58i)15-s + 4.48·16-s + (−0.850 − 0.850i)17-s + (−3.15 − 3.26i)18-s + 6.87i·19-s + ⋯
L(s)  = 1  + (−0.755 − 0.755i)2-s + (0.999 + 0.00867i)3-s + 0.142i·4-s + (−0.908 + 0.417i)5-s + (−0.749 − 0.762i)6-s + (−0.648 + 0.648i)8-s + (0.999 + 0.0173i)9-s + (1.00 + 0.371i)10-s + 0.275i·11-s + (−0.00123 + 0.142i)12-s + (0.854 + 0.854i)13-s + (−0.912 + 0.409i)15-s + 1.12·16-s + (−0.206 − 0.206i)17-s + (−0.742 − 0.768i)18-s + 1.57i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.991 + 0.131i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.991 + 0.131i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20148 - 0.0792645i\)
\(L(\frac12)\) \(\approx\) \(1.20148 - 0.0792645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0150i)T \)
5 \( 1 + (2.03 - 0.933i)T \)
7 \( 1 \)
good2 \( 1 + (1.06 + 1.06i)T + 2iT^{2} \)
11 \( 1 - 0.914iT - 11T^{2} \)
13 \( 1 + (-3.07 - 3.07i)T + 13iT^{2} \)
17 \( 1 + (0.850 + 0.850i)T + 17iT^{2} \)
19 \( 1 - 6.87iT - 19T^{2} \)
23 \( 1 + (-1.38 + 1.38i)T - 23iT^{2} \)
29 \( 1 - 2.72T + 29T^{2} \)
31 \( 1 - 4.63T + 31T^{2} \)
37 \( 1 + (-0.567 + 0.567i)T - 37iT^{2} \)
41 \( 1 + 0.922iT - 41T^{2} \)
43 \( 1 + (4.80 + 4.80i)T + 43iT^{2} \)
47 \( 1 + (-7.41 - 7.41i)T + 47iT^{2} \)
53 \( 1 + (7.79 - 7.79i)T - 53iT^{2} \)
59 \( 1 - 9.88T + 59T^{2} \)
61 \( 1 + 1.06T + 61T^{2} \)
67 \( 1 + (5.00 - 5.00i)T - 67iT^{2} \)
71 \( 1 + 0.557iT - 71T^{2} \)
73 \( 1 + (1.54 + 1.54i)T + 73iT^{2} \)
79 \( 1 - 3.03iT - 79T^{2} \)
83 \( 1 + (2.38 - 2.38i)T - 83iT^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (1.58 - 1.58i)T - 97iT^{2} \)
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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

−10.32690941640510212992374643083, −9.536654234144507404850538246095, −8.650591919326204567107019902108, −8.167817612616223164980060161456, −7.17606537361016993346455733045, −6.14254266299575958293230330588, −4.47250715141950208895915106183, −3.56230712420444956683750975062, −2.54146636882270744756775276928, −1.33944347572669562837740116739, 0.825493126417572290331265355544, 2.93791156909308491874871419201, 3.73181083740102718210660757466, 4.89602956437206272104465127341, 6.41733434939839345420690530675, 7.22583795708966577425799566253, 8.027715112617519271159118679212, 8.551641487874440809920099936653, 9.071433464692570245038433725758, 10.07596966516168998167257019736

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