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Properties

Label 2-1400-35.9-c1-0-1
Degree $2$
Conductor $1400$
Sign $-0.932 + 0.362i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
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  + (−0.611 + 0.352i)3-s + (−2.56 + 0.647i)7-s + (−1.25 + 2.16i)9-s + (2.25 + 3.89i)11-s − 5.09i·13-s + (1.73 − i)17-s + (−1.54 + 2.67i)19-s + (1.33 − 1.30i)21-s + (5.01 + 2.89i)23-s − 3.88i·27-s − 9.50·29-s + (−2.70 − 4.68i)31-s + (−2.75 − 1.58i)33-s + (−6.14 − 3.54i)37-s + (1.79 + 3.11i)39-s + ⋯
L(s)  = 1  + (−0.352 + 0.203i)3-s + (−0.969 + 0.244i)7-s + (−0.416 + 0.722i)9-s + (0.678 + 1.17i)11-s − 1.41i·13-s + (0.420 − 0.242i)17-s + (−0.354 + 0.614i)19-s + (0.292 − 0.283i)21-s + (1.04 + 0.604i)23-s − 0.747i·27-s − 1.76·29-s + (−0.485 − 0.841i)31-s + (−0.478 − 0.276i)33-s + (−1.00 − 0.582i)37-s + (0.287 + 0.498i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.932 + 0.362i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.932 + 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09738818160\)
\(L(\frac12)\) \(\approx\) \(0.09738818160\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.56 - 0.647i)T \)
good3 \( 1 + (0.611 - 0.352i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.25 - 3.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.09iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.54 - 2.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.01 - 2.89i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.50T + 29T^{2} \)
31 \( 1 + (2.70 + 4.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.14 + 3.54i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.59T + 41T^{2} \)
43 \( 1 + 4.70iT - 43T^{2} \)
47 \( 1 + (8.74 + 5.04i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.58 - 4.95i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.45 + 7.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.101 - 0.0585i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (7.08 - 4.09i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.09 - 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.7iT - 83T^{2} \)
89 \( 1 + (-2.04 + 3.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−9.933058047899096452120543308267, −9.432820884191245212946804211923, −8.415664372591455484980410939924, −7.49726449805830878346183358461, −6.81736611385803215924524710984, −5.60587932316642028074811571368, −5.32736601838839086961036825025, −3.96946105413341868397512361281, −3.09663109992815573027541758746, −1.86264079999692091780253801813, 0.04106159755717633595805139033, 1.45874134820228445734771226159, 3.16162950836861070120587576925, 3.69147814647726247269686742559, 4.95223840719285374994521404828, 6.09532557530213994328890150823, 6.56091725046599390422684806504, 7.16978689436371232073653647397, 8.630609243206225281932835224958, 9.039404337681761833432388073243

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