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Properties

Label 2-945-105.62-c0-0-3
Degree $2$
Conductor $945$
Sign $0.525 + 0.850i$
Analytic cond. $0.471616$
Root an. cond. $0.686743$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (0.707 − 0.707i)5-s − i·7-s + (0.707 + 0.707i)8-s − 1.00i·10-s + 1.41i·11-s + (−1 − i)13-s + (−0.707 − 0.707i)14-s + 1.00·16-s + (0.707 + 0.707i)17-s − 19-s + (1.00 + 1.00i)22-s + (−0.707 − 0.707i)23-s − 1.00i·25-s − 1.41·26-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.707 − 0.707i)5-s − i·7-s + (0.707 + 0.707i)8-s − 1.00i·10-s + 1.41i·11-s + (−1 − i)13-s + (−0.707 − 0.707i)14-s + 1.00·16-s + (0.707 + 0.707i)17-s − 19-s + (1.00 + 1.00i)22-s + (−0.707 − 0.707i)23-s − 1.00i·25-s − 1.41·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(0.471616\)
Root analytic conductor: \(0.686743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.584836162\)
\(L(\frac12)\) \(\approx\) \(1.584836162\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
good2 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + iT - T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−10.32710541285628860483510501534, −9.631154873229329999296751049732, −8.310873050384492199329734568634, −7.69689996313707554184181852290, −6.67978249173192136836892297664, −5.38723808845319885427882355858, −4.64482083401901467967038542011, −3.97088247098110618627939855363, −2.61435466688517296327359667403, −1.61768374232197048883627243899, 1.98095296064903022694035802429, 3.10217126675656477775142127368, 4.37870389741006193290514274149, 5.58443390883765551804659673178, 5.86191421297918258788169360489, 6.73897059620165097675706360139, 7.57459240997438400057108936124, 8.748928231997130367610325222358, 9.608128699527265909654757361245, 10.23337471090232716034278747351

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