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Properties

Label 2-1935-5.4-c1-0-102
Degree $2$
Conductor $1935$
Sign $0.0935 - 0.995i$
Analytic cond. $15.4510$
Root an. cond. $3.93078$
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  − 2.19i·2-s − 2.79·4-s + (2.22 + 0.209i)5-s − 4.99i·7-s + 1.74i·8-s + (0.458 − 4.87i)10-s − 5.42·11-s + 5.08i·13-s − 10.9·14-s − 1.77·16-s − 2.09i·17-s − 7.09·19-s + (−6.22 − 0.584i)20-s + 11.8i·22-s + 1.86i·23-s + ⋯
L(s)  = 1  − 1.54i·2-s − 1.39·4-s + (0.995 + 0.0935i)5-s − 1.88i·7-s + 0.616i·8-s + (0.144 − 1.54i)10-s − 1.63·11-s + 1.41i·13-s − 2.92·14-s − 0.443·16-s − 0.508i·17-s − 1.62·19-s + (−1.39 − 0.130i)20-s + 2.53i·22-s + 0.388i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0935 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0935 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1935\)    =    \(3^{2} \cdot 5 \cdot 43\)
Sign: $0.0935 - 0.995i$
Analytic conductor: \(15.4510\)
Root analytic conductor: \(3.93078\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1935} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1935,\ (\ :1/2),\ 0.0935 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7084592989\)
\(L(\frac12)\) \(\approx\) \(0.7084592989\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.209i)T \)
43 \( 1 + iT \)
good2 \( 1 + 2.19iT - 2T^{2} \)
7 \( 1 + 4.99iT - 7T^{2} \)
11 \( 1 + 5.42T + 11T^{2} \)
13 \( 1 - 5.08iT - 13T^{2} \)
17 \( 1 + 2.09iT - 17T^{2} \)
19 \( 1 + 7.09T + 19T^{2} \)
23 \( 1 - 1.86iT - 23T^{2} \)
29 \( 1 - 1.21T + 29T^{2} \)
31 \( 1 + 3.60T + 31T^{2} \)
37 \( 1 - 0.533iT - 37T^{2} \)
41 \( 1 - 8.69T + 41T^{2} \)
47 \( 1 - 1.27iT - 47T^{2} \)
53 \( 1 + 6.46iT - 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 - 6.49T + 71T^{2} \)
73 \( 1 + 0.220iT - 73T^{2} \)
79 \( 1 + 4.51T + 79T^{2} \)
83 \( 1 + 9.92iT - 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 9.19iT - 97T^{2} \)
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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

−9.072298843309831392034168965573, −7.86193009357731573541992941376, −7.03493249132511400097412160252, −6.25611940128409454164564225320, −4.84005132224541604133874048544, −4.36693554131562998534824760885, −3.35303653795173759997368389277, −2.32916672278109568421955867871, −1.57552291207756853437123045737, −0.23275087277723589799925284925, 2.23405673151101877957071831020, 2.77773623588519162378469198421, 4.71034702287639808791586243426, 5.35878381180688457407827793316, 5.94202386092525060481308038711, 6.22494871483469770027160452565, 7.54240427183438435036183456359, 8.257265463882297312832726440243, 8.691898685602636745613192053027, 9.405560324969131553441856239360

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