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Properties

Label 2-1408-176.131-c1-0-14
Degree $2$
Conductor $1408$
Sign $0.469 - 0.883i$
Analytic cond. $11.2429$
Root an. cond. $3.35304$
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  + (0.429 − 0.429i)3-s + (−0.703 + 0.703i)5-s − 3.48i·7-s + 2.63i·9-s + (−2.86 + 1.67i)11-s + (−0.438 + 0.438i)13-s + 0.604i·15-s + 3.89i·17-s + (0.685 + 0.685i)19-s + (−1.49 − 1.49i)21-s + 3.54·23-s + 4.00i·25-s + (2.42 + 2.42i)27-s + (3.67 − 3.67i)29-s + 8.85i·31-s + ⋯
L(s)  = 1  + (0.248 − 0.248i)3-s + (−0.314 + 0.314i)5-s − 1.31i·7-s + 0.876i·9-s + (−0.863 + 0.504i)11-s + (−0.121 + 0.121i)13-s + 0.156i·15-s + 0.945i·17-s + (0.157 + 0.157i)19-s + (−0.327 − 0.327i)21-s + 0.738·23-s + 0.801i·25-s + (0.465 + 0.465i)27-s + (0.682 − 0.682i)29-s + 1.59i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $0.469 - 0.883i$
Analytic conductor: \(11.2429\)
Root analytic conductor: \(3.35304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1408} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1408,\ (\ :1/2),\ 0.469 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.355877923\)
\(L(\frac12)\) \(\approx\) \(1.355877923\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (2.86 - 1.67i)T \)
good3 \( 1 + (-0.429 + 0.429i)T - 3iT^{2} \)
5 \( 1 + (0.703 - 0.703i)T - 5iT^{2} \)
7 \( 1 + 3.48iT - 7T^{2} \)
13 \( 1 + (0.438 - 0.438i)T - 13iT^{2} \)
17 \( 1 - 3.89iT - 17T^{2} \)
19 \( 1 + (-0.685 - 0.685i)T + 19iT^{2} \)
23 \( 1 - 3.54T + 23T^{2} \)
29 \( 1 + (-3.67 + 3.67i)T - 29iT^{2} \)
31 \( 1 - 8.85iT - 31T^{2} \)
37 \( 1 + (4.70 - 4.70i)T - 37iT^{2} \)
41 \( 1 - 6.55T + 41T^{2} \)
43 \( 1 + (-4.01 + 4.01i)T - 43iT^{2} \)
47 \( 1 + 5.68iT - 47T^{2} \)
53 \( 1 + (6.67 - 6.67i)T - 53iT^{2} \)
59 \( 1 + (-10.5 - 10.5i)T + 59iT^{2} \)
61 \( 1 + (-3.64 + 3.64i)T - 61iT^{2} \)
67 \( 1 + (-3.52 + 3.52i)T - 67iT^{2} \)
71 \( 1 - 6.71T + 71T^{2} \)
73 \( 1 + 8.28T + 73T^{2} \)
79 \( 1 + 8.89T + 79T^{2} \)
83 \( 1 + (1.57 + 1.57i)T + 83iT^{2} \)
89 \( 1 - 8.43iT - 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−9.926165350339409808114121186674, −8.697312462812899565608113973168, −7.925902690470603117252967050389, −7.30841173659336206559827248869, −6.78337806862898499335338852254, −5.41234214501881257056547965541, −4.59192231227731302955703369999, −3.65133992071492611975214472692, −2.58965011822726046717471110436, −1.32114481915299249740277543685, 0.56688207756559892834804482082, 2.47751451705563515068705271215, 3.10517784529053592197626607822, 4.34782492240131076203204760393, 5.28970365064096576320353993984, 5.96437102397663329264276823615, 6.99546537090701463079434638534, 8.039061721827891754783742719286, 8.678641543682463741879158808137, 9.322145427591273239423172447986

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