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Properties

Label 2-3234-77.76-c1-0-24
Degree $2$
Conductor $3234$
Sign $0.895 + 0.444i$
Analytic cond. $25.8236$
Root an. cond. $5.08169$
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  − i·2-s − i·3-s − 4-s + 0.357i·5-s − 6-s + i·8-s − 9-s + 0.357·10-s + (−3.31 − 0.129i)11-s + i·12-s + 4.33·13-s + 0.357·15-s + 16-s + 0.347·17-s + i·18-s − 7.81·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.159i·5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s + 0.112·10-s + (−0.999 − 0.0389i)11-s + 0.288i·12-s + 1.20·13-s + 0.0922·15-s + 0.250·16-s + 0.0842·17-s + 0.235i·18-s − 1.79·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(25.8236\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3234} (2155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3234,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.420105821\)
\(L(\frac12)\) \(\approx\) \(1.420105821\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + iT \)
7 \( 1 \)
11 \( 1 + (3.31 + 0.129i)T \)
good5 \( 1 - 0.357iT - 5T^{2} \)
13 \( 1 - 4.33T + 13T^{2} \)
17 \( 1 - 0.347T + 17T^{2} \)
19 \( 1 + 7.81T + 19T^{2} \)
23 \( 1 - 6.36T + 23T^{2} \)
29 \( 1 - 7.67iT - 29T^{2} \)
31 \( 1 - 6.20iT - 31T^{2} \)
37 \( 1 + 8.95T + 37T^{2} \)
41 \( 1 - 5.43T + 41T^{2} \)
43 \( 1 + 2.91iT - 43T^{2} \)
47 \( 1 - 3.84iT - 47T^{2} \)
53 \( 1 + 2.73T + 53T^{2} \)
59 \( 1 - 0.911iT - 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 - 6.18T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 + 6.65iT - 79T^{2} \)
83 \( 1 - 16.7T + 83T^{2} \)
89 \( 1 - 14.5iT - 89T^{2} \)
97 \( 1 - 18.1iT - 97T^{2} \)
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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

−8.702259022823307434835539979507, −8.064227875662871787813604276052, −7.00305715782183412778117191079, −6.49160333768962290806515394865, −5.42145990953975656600194152100, −4.79642263853537133682589387974, −3.61020019494911516573722509267, −2.92733412803787526943369013591, −1.95819305447583390514405833697, −0.932623548264701062941193283024, 0.55201754387237287577678545830, 2.21284641511540281032999500538, 3.33221108556385214202294560127, 4.23381799846769837250319192805, 4.88340163478589289773854788825, 5.73260944999893797505651931949, 6.35505191686144383510677785121, 7.18918348330808682570808292445, 8.203346628851156705991816816026, 8.525075481833505927595164386910

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