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Properties

Label 2-1412-1412.1143-c0-0-0
Degree $2$
Conductor $1412$
Sign $0.869 - 0.493i$
Analytic cond. $0.704679$
Root an. cond. $0.839452$
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  + (0.281 + 0.959i)2-s + (−0.841 + 0.540i)4-s + (1.07 − 1.33i)5-s + (−0.755 − 0.654i)8-s + (0.212 + 0.977i)9-s + (1.58 + 0.658i)10-s + (1.45 − 1.35i)13-s + (0.415 − 0.909i)16-s + (−1.27 − 1.10i)17-s + (−0.877 + 0.479i)18-s + (−0.183 + 1.71i)20-s + (−0.416 − 1.91i)25-s + (1.71 + 1.01i)26-s + (−1.01 + 1.57i)29-s + (0.989 + 0.142i)32-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)2-s + (−0.841 + 0.540i)4-s + (1.07 − 1.33i)5-s + (−0.755 − 0.654i)8-s + (0.212 + 0.977i)9-s + (1.58 + 0.658i)10-s + (1.45 − 1.35i)13-s + (0.415 − 0.909i)16-s + (−1.27 − 1.10i)17-s + (−0.877 + 0.479i)18-s + (−0.183 + 1.71i)20-s + (−0.416 − 1.91i)25-s + (1.71 + 1.01i)26-s + (−1.01 + 1.57i)29-s + (0.989 + 0.142i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $0.869 - 0.493i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (1143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ 0.869 - 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.388634546\)
\(L(\frac12)\) \(\approx\) \(1.388634546\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.281 - 0.959i)T \)
353 \( 1 + (0.654 + 0.755i)T \)
good3 \( 1 + (-0.212 - 0.977i)T^{2} \)
5 \( 1 + (-1.07 + 1.33i)T + (-0.212 - 0.977i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.959 + 0.281i)T^{2} \)
13 \( 1 + (-1.45 + 1.35i)T + (0.0713 - 0.997i)T^{2} \)
17 \( 1 + (1.27 + 1.10i)T + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (-0.755 - 0.654i)T^{2} \)
23 \( 1 + (0.281 - 0.959i)T^{2} \)
29 \( 1 + (1.01 - 1.57i)T + (-0.415 - 0.909i)T^{2} \)
31 \( 1 + (-0.877 - 0.479i)T^{2} \)
37 \( 1 + (-0.599 - 0.199i)T + (0.800 + 0.599i)T^{2} \)
41 \( 1 + (-1.04 - 0.784i)T + (0.281 + 0.959i)T^{2} \)
43 \( 1 + (-0.281 - 0.959i)T^{2} \)
47 \( 1 + (-0.909 - 0.415i)T^{2} \)
53 \( 1 + (-0.175 - 1.63i)T + (-0.977 + 0.212i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.635 - 0.290i)T + (0.654 + 0.755i)T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + (0.349 + 0.936i)T^{2} \)
73 \( 1 + (0.764 + 1.18i)T + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.212 - 0.977i)T^{2} \)
83 \( 1 + (-0.142 + 0.989i)T^{2} \)
89 \( 1 + (-0.322 - 0.968i)T + (-0.800 + 0.599i)T^{2} \)
97 \( 1 + (1.25 + 1.45i)T + (-0.142 + 0.989i)T^{2} \)
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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

−9.402605804247138981490959335479, −8.944528989210690897609550304447, −8.230568450752523589713165042895, −7.46849046965379615532445735751, −6.31614430935378122077563075272, −5.59981895336460233872378223451, −5.02905066927760375890942433732, −4.26518512246028595230489508937, −2.81077996968764203052016829016, −1.25842026619676697531572887096, 1.68590942534322884244445479333, 2.39563648179497330084137375833, 3.70860836238089633783056407906, 4.10578429429620220957496297636, 5.76362764590255984013175540537, 6.29998860956212848845850525272, 6.83268778281132278329259834313, 8.418698730994571467963332613895, 9.255030979109586765007350208927, 9.716943390316107442963867791426

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