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Properties

Label 2-936-104.99-c1-0-46
Degree $2$
Conductor $936$
Sign $0.899 - 0.437i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
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  + (−0.437 + 1.34i)2-s + (−1.61 − 1.17i)4-s + (1.98 − 1.98i)5-s + (3.05 + 3.05i)7-s + (2.28 − 1.66i)8-s + (1.79 + 3.52i)10-s + (1.08 − 1.08i)11-s + (3.57 − 0.429i)13-s + (−5.45 + 2.77i)14-s + (1.23 + 3.80i)16-s − 5.70i·17-s + (−4.39 − 4.39i)19-s + (−5.53 + 0.874i)20-s + (0.984 + 1.93i)22-s − 2.95·23-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.808 − 0.588i)4-s + (0.885 − 0.885i)5-s + (1.15 + 1.15i)7-s + (0.809 − 0.587i)8-s + (0.568 + 1.11i)10-s + (0.327 − 0.327i)11-s + (0.992 − 0.119i)13-s + (−1.45 + 0.742i)14-s + (0.308 + 0.951i)16-s − 1.38i·17-s + (−1.00 − 1.00i)19-s + (−1.23 + 0.195i)20-s + (0.210 + 0.412i)22-s − 0.615·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.899 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67064 + 0.385154i\)
\(L(\frac12)\) \(\approx\) \(1.67064 + 0.385154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.437 - 1.34i)T \)
3 \( 1 \)
13 \( 1 + (-3.57 + 0.429i)T \)
good5 \( 1 + (-1.98 + 1.98i)T - 5iT^{2} \)
7 \( 1 + (-3.05 - 3.05i)T + 7iT^{2} \)
11 \( 1 + (-1.08 + 1.08i)T - 11iT^{2} \)
17 \( 1 + 5.70iT - 17T^{2} \)
19 \( 1 + (4.39 + 4.39i)T + 19iT^{2} \)
23 \( 1 + 2.95T + 23T^{2} \)
29 \( 1 + 6.96iT - 29T^{2} \)
31 \( 1 + (3.05 - 3.05i)T - 31iT^{2} \)
37 \( 1 + (-5.14 - 5.14i)T + 37iT^{2} \)
41 \( 1 + (-2.77 - 2.77i)T + 41iT^{2} \)
43 \( 1 + 3.00iT - 43T^{2} \)
47 \( 1 + (-6.40 - 6.40i)T + 47iT^{2} \)
53 \( 1 + 4.28iT - 53T^{2} \)
59 \( 1 + (3.00 - 3.00i)T - 59iT^{2} \)
61 \( 1 + 1.13iT - 61T^{2} \)
67 \( 1 + (-7.39 - 7.39i)T + 67iT^{2} \)
71 \( 1 + (3.20 - 3.20i)T - 71iT^{2} \)
73 \( 1 + (-7.87 + 7.87i)T - 73iT^{2} \)
79 \( 1 + 4.46iT - 79T^{2} \)
83 \( 1 + (-9.78 - 9.78i)T + 83iT^{2} \)
89 \( 1 + (4.24 - 4.24i)T - 89iT^{2} \)
97 \( 1 + (11.5 + 11.5i)T + 97iT^{2} \)
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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

−9.648240625352031811952995521603, −9.064017589106509462595742725447, −8.552954090051357482303223752431, −7.83551067229927828200855372336, −6.50686677860678854240708161085, −5.77489002567125203090788314168, −5.12091264690672800717637268799, −4.34152680761586483963483095344, −2.31276783553083929513235215031, −1.08203166894260974917670063878, 1.44044850450930243413598088173, 2.10881643333871998360191422878, 3.76000113467158401289081312931, 4.18758920866848224684433403433, 5.64294473489570591088701161042, 6.61880367849071335547404396036, 7.72019005264307133496643713070, 8.368094432422201313003687807549, 9.366625975318979878693992674289, 10.47280971832173532117465218632

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