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Properties

Label 2-1078-77.76-c1-0-25
Degree $2$
Conductor $1078$
Sign $-0.337 + 0.941i$
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  − i·2-s − 1.55i·3-s − 4-s + 1.01i·5-s − 1.55·6-s + i·8-s + 0.568·9-s + 1.01·10-s + (3.09 − 1.19i)11-s + 1.55i·12-s − 0.167·13-s + 1.58·15-s + 16-s + 2.95·17-s − 0.568i·18-s + 0.311·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.900i·3-s − 0.5·4-s + 0.455i·5-s − 0.636·6-s + 0.353i·8-s + 0.189·9-s + 0.322·10-s + (0.932 − 0.360i)11-s + 0.450i·12-s − 0.0463·13-s + 0.410·15-s + 0.250·16-s + 0.716·17-s − 0.134i·18-s + 0.0715·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.337 + 0.941i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.337 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.684150703\)
\(L(\frac12)\) \(\approx\) \(1.684150703\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
7 \( 1 \)
11 \( 1 + (-3.09 + 1.19i)T \)
good3 \( 1 + 1.55iT - 3T^{2} \)
5 \( 1 - 1.01iT - 5T^{2} \)
13 \( 1 + 0.167T + 13T^{2} \)
17 \( 1 - 2.95T + 17T^{2} \)
19 \( 1 - 0.311T + 19T^{2} \)
23 \( 1 + 0.475T + 23T^{2} \)
29 \( 1 + 1.89iT - 29T^{2} \)
31 \( 1 - 2.54iT - 31T^{2} \)
37 \( 1 - 6.09T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + 3.79iT - 47T^{2} \)
53 \( 1 - 8.42T + 53T^{2} \)
59 \( 1 + 6.24iT - 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 - 9.70T + 73T^{2} \)
79 \( 1 - 7.00iT - 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 6.90iT - 89T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−9.771709209507222414931677771456, −8.824141667888457746240999467723, −8.001328611016199355984901218103, −7.05409140125311187094139506455, −6.43965984044220764784561086761, −5.33362446737946647576886969129, −4.09140031784221112396208060259, −3.16659272672029409830071678061, −1.98487973423324115163027718469, −0.923038036608626840282288375545, 1.30767683655954480293294324587, 3.25066783825006565837053531662, 4.28330608145572269565151663612, 4.82186345872923114808837655160, 5.82200301440321180948950487453, 6.77919658392742081257189119105, 7.62974815407451013713163831222, 8.598760024633027752034982953220, 9.354124581153818896057820703371, 9.845907736878387326398985302198

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