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Properties

Label 2-1078-77.10-c1-0-31
Degree $2$
Conductor $1078$
Sign $0.999 + 0.00588i$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
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  + (0.866 − 0.5i)2-s + (2.70 + 1.56i)3-s + (0.499 − 0.866i)4-s + (1.90 − 1.10i)5-s + 3.12·6-s − 0.999i·8-s + (3.38 + 5.87i)9-s + (1.10 − 1.90i)10-s + (−1.82 − 2.77i)11-s + (2.70 − 1.56i)12-s + 1.45·13-s + 6.88·15-s + (−0.5 − 0.866i)16-s + (−3.80 + 6.58i)17-s + (5.87 + 3.38i)18-s + (−0.0903 − 0.156i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (1.56 + 0.902i)3-s + (0.249 − 0.433i)4-s + (0.853 − 0.492i)5-s + 1.27·6-s − 0.353i·8-s + (1.12 + 1.95i)9-s + (0.348 − 0.603i)10-s + (−0.548 − 0.835i)11-s + (0.781 − 0.451i)12-s + 0.404·13-s + 1.77·15-s + (−0.125 − 0.216i)16-s + (−0.922 + 1.59i)17-s + (1.38 + 0.798i)18-s + (−0.0207 − 0.0358i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.999 + 0.00588i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.999 + 0.00588i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.239597092\)
\(L(\frac12)\) \(\approx\) \(4.239597092\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 \)
11 \( 1 + (1.82 + 2.77i)T \)
good3 \( 1 + (-2.70 - 1.56i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.90 + 1.10i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.45T + 13T^{2} \)
17 \( 1 + (3.80 - 6.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0903 + 0.156i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.14 + 1.98i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.45iT - 29T^{2} \)
31 \( 1 + (7.40 + 4.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.754 + 1.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.10T + 41T^{2} \)
43 \( 1 + 1.58iT - 43T^{2} \)
47 \( 1 + (-0.472 + 0.272i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.41 - 4.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.36 - 3.09i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.86 + 4.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.49 + 2.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.57T + 71T^{2} \)
73 \( 1 + (4.83 - 8.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.99 + 3.45i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.84T + 83T^{2} \)
89 \( 1 + (13.9 - 8.05i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.3iT - 97T^{2} \)
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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

−9.809694085280895022445618061640, −9.081632167112400437538215382913, −8.512619668580655757318147621790, −7.69569685143090875483050302133, −6.18148472129568177116642805868, −5.44554785922387390970470101255, −4.28183056547471768202619122425, −3.72703648686431479263592066811, −2.58274977632026422659757661831, −1.81479321317165929786873535885, 1.79581537253154584115322457438, 2.54495810305820921863869443528, 3.32596373871900368470744456748, 4.58208271331148095974281882278, 5.76039441497734176008313108103, 6.92622315902362153673201572903, 7.13228224476402848247743441280, 8.061667200368956607410748238055, 9.030388600701722032396116772148, 9.540129260450058862654135333644

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