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Properties

Label 2-308-11.9-c1-0-2
Degree $2$
Conductor $308$
Sign $0.987 + 0.157i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
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  + (−2.11 + 1.53i)3-s + (−0.864 − 2.65i)5-s + (0.809 + 0.587i)7-s + (1.19 − 3.66i)9-s + (2.88 + 1.64i)11-s + (1.55 − 4.78i)13-s + (5.92 + 4.30i)15-s + (1.15 + 3.54i)17-s + (2.85 − 2.07i)19-s − 2.61·21-s + 4.61·23-s + (−2.28 + 1.65i)25-s + (0.690 + 2.12i)27-s + (6.58 + 4.78i)29-s + (1.84 − 5.67i)31-s + ⋯
L(s)  = 1  + (−1.22 + 0.888i)3-s + (−0.386 − 1.18i)5-s + (0.305 + 0.222i)7-s + (0.396 − 1.22i)9-s + (0.868 + 0.495i)11-s + (0.431 − 1.32i)13-s + (1.52 + 1.11i)15-s + (0.279 + 0.860i)17-s + (0.654 − 0.475i)19-s − 0.571·21-s + 0.962·23-s + (−0.456 + 0.331i)25-s + (0.132 + 0.409i)27-s + (1.22 + 0.888i)29-s + (0.331 − 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.987 + 0.157i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.987 + 0.157i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886180 - 0.0704436i\)
\(L(\frac12)\) \(\approx\) \(0.886180 - 0.0704436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-2.88 - 1.64i)T \)
good3 \( 1 + (2.11 - 1.53i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.864 + 2.65i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.55 + 4.78i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.15 - 3.54i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.85 + 2.07i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 + (-6.58 - 4.78i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.84 + 5.67i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.73 + 4.16i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.11 + 1.53i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.77T + 43T^{2} \)
47 \( 1 + (6.67 - 4.85i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.25 + 13.0i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.58 + 2.60i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.90 + 5.87i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 0.753T + 67T^{2} \)
71 \( 1 + (-3.87 - 11.9i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.48 + 1.80i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.859 + 2.64i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.39 - 10.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 4.61T + 89T^{2} \)
97 \( 1 + (4.73 - 14.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.65766274035157783792902879832, −10.82744399753994874755463384464, −9.931970731621581337908334450654, −8.939570133973036893228999691686, −8.041464878304081064637201049524, −6.51603465551227444621656211384, −5.32785678709762189882117387838, −4.84660825101381793189981299151, −3.68480467400602413218089415321, −0.962464030653274753260292506333, 1.29505426949870086535194203520, 3.24505943625167527972377043806, 4.76135475381809866907942031659, 6.16355882917051441053687463940, 6.79338202302568719371234251930, 7.41003892699196070994128243137, 8.808786223446450629052740524553, 10.19983211025918487197911909492, 11.17014248108352296208526674295, 11.66537020924289648776897898295

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