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Properties

Label 2-26e2-52.7-c1-0-30
Degree $2$
Conductor $676$
Sign $0.846 + 0.533i$
Analytic cond. $5.39788$
Root an. cond. $2.32333$
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (0.633 + 0.633i)5-s + (−1.99 + 2i)8-s + (−1.5 − 2.59i)9-s + (−1.09 − 0.633i)10-s + (1.99 − 3.46i)16-s + (6.86 − 3.96i)17-s + (3 + 3i)18-s + (1.73 + 0.464i)20-s − 4.19i·25-s + (−3.33 + 5.76i)29-s + (−1.46 + 5.46i)32-s + (−7.92 + 7.92i)34-s + (−5.19 − 3i)36-s + (−3.13 − 11.6i)37-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (0.283 + 0.283i)5-s + (−0.707 + 0.707i)8-s + (−0.5 − 0.866i)9-s + (−0.347 − 0.200i)10-s + (0.499 − 0.866i)16-s + (1.66 − 0.961i)17-s + (0.707 + 0.707i)18-s + (0.387 + 0.103i)20-s − 0.839i·25-s + (−0.618 + 1.07i)29-s + (−0.258 + 0.965i)32-s + (−1.35 + 1.35i)34-s + (−0.866 − 0.5i)36-s + (−0.515 − 1.92i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.846 + 0.533i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ 0.846 + 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923064 - 0.266537i\)
\(L(\frac12)\) \(\approx\) \(0.923064 - 0.266537i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
13 \( 1 \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.633 - 0.633i)T + 5iT^{2} \)
7 \( 1 + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-6.86 + 3.96i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.33 - 5.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 31iT^{2} \)
37 \( 1 + (3.13 + 11.6i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-9.96 + 2.66i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.69 + 4.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.83 + 9.83i)T - 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-1.09 - 4.09i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.83 + 6.83i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.30317985145771213839658726220, −9.396139522412526130790625498783, −8.921113048777513909476075670221, −7.74050963857889941681364208231, −7.08974936897077902993876912209, −6.02928146702536396973022748388, −5.38170768519088933628423455010, −3.54551867325698144782463907228, −2.44079524416668977522201047399, −0.78551612944828513337145086083, 1.31485171859850639601161700630, 2.55952379228309053592590388128, 3.76844635330850494666538306762, 5.34890766512252320195079206105, 6.12523187737391858692308823659, 7.43850831718972994453358134870, 8.051397596873795694761082867750, 8.826117396074277961481511782011, 9.832026810634489408890965357247, 10.36852236584503445338688809247

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