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Properties

Label 2-945-15.2-c1-0-3
Degree $2$
Conductor $945$
Sign $-0.794 + 0.607i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + (−1.80 + 1.80i)2-s − 4.51i·4-s + (−2.22 + 0.221i)5-s + (−0.707 − 0.707i)7-s + (4.53 + 4.53i)8-s + (3.61 − 4.41i)10-s + 2.89i·11-s + (2.10 − 2.10i)13-s + 2.55·14-s − 7.34·16-s + (2.77 − 2.77i)17-s + 7.55i·19-s + (1.00 + 10.0i)20-s + (−5.22 − 5.22i)22-s + (−1.24 − 1.24i)23-s + ⋯
L(s)  = 1  + (−1.27 + 1.27i)2-s − 2.25i·4-s + (−0.995 + 0.0992i)5-s + (−0.267 − 0.267i)7-s + (1.60 + 1.60i)8-s + (1.14 − 1.39i)10-s + 0.873i·11-s + (0.583 − 0.583i)13-s + 0.682·14-s − 1.83·16-s + (0.673 − 0.673i)17-s + 1.73i·19-s + (0.223 + 2.24i)20-s + (−1.11 − 1.11i)22-s + (−0.258 − 0.258i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.794 + 0.607i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0644889 - 0.190464i\)
\(L(\frac12)\) \(\approx\) \(0.0644889 - 0.190464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.22 - 0.221i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (1.80 - 1.80i)T - 2iT^{2} \)
11 \( 1 - 2.89iT - 11T^{2} \)
13 \( 1 + (-2.10 + 2.10i)T - 13iT^{2} \)
17 \( 1 + (-2.77 + 2.77i)T - 17iT^{2} \)
19 \( 1 - 7.55iT - 19T^{2} \)
23 \( 1 + (1.24 + 1.24i)T + 23iT^{2} \)
29 \( 1 - 2.36T + 29T^{2} \)
31 \( 1 - 2.63T + 31T^{2} \)
37 \( 1 + (8.24 + 8.24i)T + 37iT^{2} \)
41 \( 1 - 1.28iT - 41T^{2} \)
43 \( 1 + (5.91 - 5.91i)T - 43iT^{2} \)
47 \( 1 + (5.81 - 5.81i)T - 47iT^{2} \)
53 \( 1 + (3.59 + 3.59i)T + 53iT^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + (-8.26 - 8.26i)T + 67iT^{2} \)
71 \( 1 + 0.883iT - 71T^{2} \)
73 \( 1 + (4.68 - 4.68i)T - 73iT^{2} \)
79 \( 1 - 13.7iT - 79T^{2} \)
83 \( 1 + (9.73 + 9.73i)T + 83iT^{2} \)
89 \( 1 + 7.33T + 89T^{2} \)
97 \( 1 + (-4.38 - 4.38i)T + 97iT^{2} \)
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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

−10.18380812181379661822346124792, −9.678170040749645873146747289269, −8.576828255620022273784376319407, −7.944016159056989425204255826874, −7.42975786134237660590735967592, −6.59422994851010012252012349325, −5.72630567121988551774260885173, −4.62482263637960531605871861964, −3.36063225608828941804142167205, −1.33552524088859807353213741104, 0.16510860537421995551429512976, 1.49194044109721745298085085879, 3.02243900647428897576824560110, 3.54415250509184602687007887871, 4.77353549653761716019889183974, 6.40155836967521546146730021009, 7.37695980861273277729146680259, 8.428126956669387864139504341884, 8.630963095430383477741426346327, 9.530424702647570653627932483812

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