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Properties

Label 2-34e2-17.13-c1-0-19
Degree $2$
Conductor $1156$
Sign $-0.992 - 0.122i$
Analytic cond. $9.23070$
Root an. cond. $3.03820$
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  + (−1.26 − 1.26i)3-s + (0.559 + 0.559i)5-s + (−3.38 + 3.38i)7-s + 0.208i·9-s + (3.24 − 3.24i)11-s + 1.79·13-s − 1.41i·15-s + 0.208i·19-s + 8.58·21-s + (−4.80 + 4.80i)23-s − 4.37i·25-s + (−3.53 + 3.53i)27-s + (−6.92 − 6.92i)29-s + (−3.53 − 3.53i)31-s − 8.20·33-s + ⋯
L(s)  = 1  + (−0.731 − 0.731i)3-s + (0.250 + 0.250i)5-s + (−1.28 + 1.28i)7-s + 0.0695i·9-s + (0.977 − 0.977i)11-s + 0.496·13-s − 0.365i·15-s + 0.0478i·19-s + 1.87·21-s + (−1.00 + 1.00i)23-s − 0.874i·25-s + (−0.680 + 0.680i)27-s + (−1.28 − 1.28i)29-s + (−0.635 − 0.635i)31-s − 1.42·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.992 - 0.122i$
Analytic conductor: \(9.23070\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :1/2),\ -0.992 - 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1405926916\)
\(L(\frac12)\) \(\approx\) \(0.1405926916\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (1.26 + 1.26i)T + 3iT^{2} \)
5 \( 1 + (-0.559 - 0.559i)T + 5iT^{2} \)
7 \( 1 + (3.38 - 3.38i)T - 7iT^{2} \)
11 \( 1 + (-3.24 + 3.24i)T - 11iT^{2} \)
13 \( 1 - 1.79T + 13T^{2} \)
19 \( 1 - 0.208iT - 19T^{2} \)
23 \( 1 + (4.80 - 4.80i)T - 23iT^{2} \)
29 \( 1 + (6.92 + 6.92i)T + 29iT^{2} \)
31 \( 1 + (3.53 + 3.53i)T + 31iT^{2} \)
37 \( 1 + (-3.94 - 3.94i)T + 37iT^{2} \)
41 \( 1 + (4.24 - 4.24i)T - 41iT^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 2.20iT - 53T^{2} \)
59 \( 1 - 9.16iT - 59T^{2} \)
61 \( 1 + (3.83 - 3.83i)T - 61iT^{2} \)
67 \( 1 + 5.58T + 67T^{2} \)
71 \( 1 + (4.24 + 4.24i)T + 71iT^{2} \)
73 \( 1 + (8.30 + 8.30i)T + 73iT^{2} \)
79 \( 1 + (1.41 - 1.41i)T - 79iT^{2} \)
83 \( 1 + 2.20iT - 83T^{2} \)
89 \( 1 + 9.16T + 89T^{2} \)
97 \( 1 + (-8.45 - 8.45i)T + 97iT^{2} \)
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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

−9.390632690530420168010708464235, −8.682649000796346082270301917820, −7.59513520197796305367889285186, −6.38544276504626552660256790823, −6.13778245544931506046987700691, −5.67463426803774353389338998736, −3.89683804355364559199979048448, −3.00620030240724282320247987777, −1.68445607394667279610990932752, −0.06624288141569739622302419792, 1.63063467078515983551882176778, 3.50683857503536937842962423548, 4.10110637650114608867908591285, 5.01528075514498445135579731295, 6.05763747819099478107754119370, 6.80425183629836159838557856728, 7.50675819134052158676089066824, 8.900572589556843323583519456876, 9.651292574010755308610560094354, 10.15031446549653984009006592655

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