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Properties

Label 2-945-15.8-c1-0-11
Degree $2$
Conductor $945$
Sign $0.984 + 0.176i$
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  + (−1.05 − 1.05i)2-s + 0.226i·4-s + (2.07 − 0.821i)5-s + (0.707 − 0.707i)7-s + (−1.87 + 1.87i)8-s + (−3.06 − 1.32i)10-s + 4.66i·11-s + (2.31 + 2.31i)13-s − 1.49·14-s + 4.40·16-s + (4.29 + 4.29i)17-s + 3.35i·19-s + (0.186 + 0.471i)20-s + (4.92 − 4.92i)22-s + (−6.03 + 6.03i)23-s + ⋯
L(s)  = 1  + (−0.746 − 0.746i)2-s + 0.113i·4-s + (0.930 − 0.367i)5-s + (0.267 − 0.267i)7-s + (−0.661 + 0.661i)8-s + (−0.968 − 0.419i)10-s + 1.40i·11-s + (0.642 + 0.642i)13-s − 0.398·14-s + 1.10·16-s + (1.04 + 1.04i)17-s + 0.770i·19-s + (0.0416 + 0.105i)20-s + (1.04 − 1.04i)22-s + (−1.25 + 1.25i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.984 + 0.176i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.984 + 0.176i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20452 - 0.107017i\)
\(L(\frac12)\) \(\approx\) \(1.20452 - 0.107017i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 + 0.821i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (1.05 + 1.05i)T + 2iT^{2} \)
11 \( 1 - 4.66iT - 11T^{2} \)
13 \( 1 + (-2.31 - 2.31i)T + 13iT^{2} \)
17 \( 1 + (-4.29 - 4.29i)T + 17iT^{2} \)
19 \( 1 - 3.35iT - 19T^{2} \)
23 \( 1 + (6.03 - 6.03i)T - 23iT^{2} \)
29 \( 1 + 1.84T + 29T^{2} \)
31 \( 1 - 6.65T + 31T^{2} \)
37 \( 1 + (4.88 - 4.88i)T - 37iT^{2} \)
41 \( 1 - 9.63iT - 41T^{2} \)
43 \( 1 + (3.88 + 3.88i)T + 43iT^{2} \)
47 \( 1 + (8.68 + 8.68i)T + 47iT^{2} \)
53 \( 1 + (0.627 - 0.627i)T - 53iT^{2} \)
59 \( 1 - 0.951T + 59T^{2} \)
61 \( 1 - 5.72T + 61T^{2} \)
67 \( 1 + (6.33 - 6.33i)T - 67iT^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 + (-0.629 - 0.629i)T + 73iT^{2} \)
79 \( 1 + 6.67iT - 79T^{2} \)
83 \( 1 + (-4.30 + 4.30i)T - 83iT^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + (-9.49 + 9.49i)T - 97iT^{2} \)
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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

−10.08170184385596335104506727406, −9.538361256716414249861599063815, −8.510974561244615306723596798312, −7.85487550593693918172782372621, −6.49711573871938702809336102797, −5.73723227995402438677732171810, −4.74817474617971026710243149696, −3.48420716484022449501820822961, −1.79000511471945162819950771921, −1.58734896381320535248670021217, 0.77123763377749977407263642941, 2.64008186631101626895721755085, 3.51337746287465761832465600851, 5.22048139256046341502788818996, 6.02205561568757685121564071022, 6.60032587253489527202036806846, 7.70590731066023645749264959937, 8.409424451651216247285088894380, 9.044913131708549891661914750852, 9.905531283436523882204406723072

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