Properties

Label 2-3234-77.76-c1-0-15
Degree $2$
Conductor $3234$
Sign $0.368 - 0.929i$
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 + 2.66i·5-s − 6-s + i·8-s − 9-s + 2.66·10-s + (−2.85 − 1.68i)11-s + i·12-s + 2.50·13-s + 2.66·15-s + 16-s + 2.53·17-s + i·18-s − 1.39·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 1.19i·5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s + 0.843·10-s + (−0.860 − 0.509i)11-s + 0.288i·12-s + 0.693·13-s + 0.688·15-s + 0.250·16-s + 0.614·17-s + 0.235i·18-s − 0.319·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $0.368 - 0.929i$
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: \((2,\ 3234,\ (\ :1/2),\ 0.368 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8227326660\)
\(L(\frac12)\) \(\approx\) \(0.8227326660\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.85 + 1.68i)T \)
good5 \( 1 - 2.66iT - 5T^{2} \)
13 \( 1 - 2.50T + 13T^{2} \)
17 \( 1 - 2.53T + 17T^{2} \)
19 \( 1 + 1.39T + 19T^{2} \)
23 \( 1 + 3.15T + 23T^{2} \)
29 \( 1 - 6.51iT - 29T^{2} \)
31 \( 1 + 4.05iT - 31T^{2} \)
37 \( 1 - 2.54T + 37T^{2} \)
41 \( 1 + 2.65T + 41T^{2} \)
43 \( 1 - 5.62iT - 43T^{2} \)
47 \( 1 - 2.76iT - 47T^{2} \)
53 \( 1 + 6.88T + 53T^{2} \)
59 \( 1 + 4.91iT - 59T^{2} \)
61 \( 1 + 5.35T + 61T^{2} \)
67 \( 1 - 7.33T + 67T^{2} \)
71 \( 1 + 6.05T + 71T^{2} \)
73 \( 1 + 9.63T + 73T^{2} \)
79 \( 1 - 8.30iT - 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 - 11.2iT - 89T^{2} \)
97 \( 1 - 4.03iT - 97T^{2} \)
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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

−8.728620072423217627919584851775, −8.013479039487000550225995559652, −7.43943581272210691037431342522, −6.45611886113140262380950004040, −5.92460925818317484441575749654, −4.97364767194818675700170053270, −3.75068883459591743342757798234, −3.06848284946776147517760142874, −2.38748326763985511943083358372, −1.23889610445185734116132462880, 0.26507333614260582099960327447, 1.65998240541909535873736187158, 3.07526295607100153449673324124, 4.19700473325200427934412775520, 4.65511154610481882195914220859, 5.52250142631129695111949310962, 5.97510762618878629421787644804, 7.10968876319423416080249467340, 7.964248222443686315356542300922, 8.466943209393057744246287767287

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